Numerical Methods for Weak Solution of Wave Equation with van der Pol Type Nonlinear Boundary Conditions Jun Liu,1 Yu Huang,2 Haiwei Sun,3 Mingqing Xiao4 1 Department of Mathematics and Statistical Sciences, Jackson State University, Jackson, Mississippi 39217 2
Department of Mathematics, Zhongshan (Sun Yat-Sen) University, Guangzhou, China
3
Department of Mathematics, University of Macau, Macao, China
4
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
Received 1 August 2014; accepted 3 June 2015 Published online 30 June 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/num.21997
We develop computational methods for solving wave equation with van der Pol type nonlinear boundary conditions under the framework of weak solutions. Based on the wave reflection on the boundaries, we first solve the Riemann invariants by constructing two iteration mappings, and then show that the weak solution can be obtained by the integration of the Riemann invariants on the boundaries. If the compatible conditions are not satisfied or only hold with a low degree, a high-order integration method is developed for the numerical solution. When the initial condition is sufficiently smooth and compatible conditions hold with a sufficient degree, we establish a sixth-order finite difference scheme, which only needs to solve a linear system at any given time instance. Numerical experiments are provided to demonstrate the effectiveness of the proposed approaches. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 373–398, 2016 Keywords: chaotic dynamics; finite difference; numerical integration; wave equation; van der Pol boundary condition; weak solution
I. INTRODUCTION
We consider the numerical solution of the one-dimensional wave equation system associated with van der Pol boundary condition in the form of
Correspondence to: Mingqing Xiao, Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA (e-mail:
[email protected]) Contract grant sponsor: National Natural Science Foundation of China; contract grant number: 11371380 Contract grant sponsor: US NSF 1021203, 1419028 © 2015 Wiley Periodicals, Inc.
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⎧ wtt − wxx = 0, x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎨w (0, t) = −ηw (0, t), η = 1, t > 0, x t 3 ⎪ (1, t) = αw (1, t) − βw (1, t), 0 < α < 1, β ≥ 0, t > 0, w x t ⎪ t ⎪ ⎩ w(x, 0) = w0 (x), wt (x, 0) = w1 (x), 0 ≤ x ≤ 1,
(1.1)
where α, β, and η are given real constants. When η = 1, the system (1.1) is not well-posed. Thus throughout this article, we assume η = 1. The wave equation itself is linear and represents the infinite-dimensional harmonic oscillator. The right-handed side boundary condition (at x = 1) is nonlinear when β = 0, which is usually called a van der Pol type boundary condition (see, e.g., [1–13]). The left-handed side boundary condition (at x = 0) is linear, where η > 0 indicates that energy is being injected into the system at x = 0. If we denote the total energy as 1 1 1 1 2 E(t) = |∇w(x, t)|2 dx = [wx (x, t) + wt2 (x, t)]dx, 2 0 2 0 then by applying the boundary conditions we have d E(t) = ηwt2 (0, t) + wt2 (1, t)[α − βwt2 (1, t)]. dt
(1.2)
Thus if η > 0, the system (1.1) has a self-excited mechanism that supplies energy to the system itself. Moreover, any constant could be an equilibrium. The PDE system (1.1) has received considerable attention since it exhibits many interesting and complicated dynamical phenomena, such as limit cycles and chaotic behavior of (wt , wx ) when the parameters α, β, and η assume certain values [1–3, 5]. Different from dynamics of a system of ODEs, this is a simple and useful infinite-dimensional model for the study of spatiotemporal behaviors as time evolutes. For instance, the propagation of acoustic waves along a cylindrical pipe satisfies the linear wave equation: wtt − wxx = 0. Its general solution is the d’Alembert solution w(x, t) = F (x − t) + G(x + t), where F , G are arbitrary functions. This solution describes a superposition of two traveling wave with arbitrary profiles, one propagating with unit speed to the left, the other with unit speed to the right. The boundary conditions appeared in (1.1) can create irregularly acoustical vibrations. This type of vibrations, for example, can be generated by noise signals radiated from underwater vehicles, and there are intensive research for the properties of acoustical vibrations in current literature (see e.g., [14] and references therein). Hence numerical study of this type of chaotic characteristics is not only important but also may lead to a better understanding of the dynamics of acoustic systems. One challenge for solving (1.1) numerically results from the fact that the wave solution of (1.1), corresponding to certain real world phenomena, is not necessary to be smooth or even to be continuous. For a given smooth initial condition (w0 , w1 ) of (1.1), the necessary and sufficient condition for (1.1) to admit a classical solution w ∈ C 2 ([0, 1] × [0, T ]) is that the initial condition (w0 , w1 ) has to satisfy the so-called compatibility conditions w0 (0) = −ηw1 (0),
w0 (1) = αw1 (1) − βw13 (1)
(1.3)
w1 (0) = −ηw0 (0),
w1 (1) = αw0 (1) − 3βw12 (1)w0 (1).
(1.4)
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If the compatibility requirement is not met, then no matter how smooth the initial conditions w0 and w1 will be, there is always a jump discontinuity in w along the characteristics of the system. This issue for general time-dependent linear initial-boundary value problems has been investigated theoretically and numerically in [15–18]. In the case of the linear wave equation with linear incompatible boundary conditions, the authors in [16] proved that a second-order finite difference method gives 4/3-order convergence and also numerically showed a fourth-order scheme achieves 3/2-order convergence. These are due to that, for hyperbolic systems, the singularities at t = 0 on the boundaries propagate throughout the domain along the characteristic lines x ± t, which presents challenges for numerical computations. For linear cases, the discontinuities usually can be accurately approximated by the discontinuous Galerkin (DG) method [19–24]. However, the applicability of DG method to our problem is challenging due to the nonlinear van der Pol type boundary condition. To this end, we emphasize that many existing high-order finite difference approaches may fail to work properly since the existence of higher derivatives is no loner to be guaranteed. Even if the necessary compatibility requirement is satisfied, the finite difference approach for (1.1) still remains challenging due to possible irregular oscillations. When parameters η and α take certain values, the gradient of w will present chaotic dynamics, which implies that in such cases the solution of (1.1) will be very sensitive to numerical errors. To illustrate it, we give the detailed description below. Denote the total variation of a function f on a closed interval I = [a, b] by n−1 VI (f ) = sup |f (xi+1 ) − f (xi )| ,
i=0
where the sup is taken over all the partition : {x0 = a < x1 < x2 < · · · < xn = b} . Let us consider the Riemann invariants of (1.1) u(x, t) =
1 (wx (x, t) + wt (x, t)), 2
v(x, t) =
1 (wx (x, t) − wt (x, t)). 2
(1.5)
Then we have the following known result that has been shown by the second author of this paper. Lemma 1 ([7]).
Let α ∈ (0, 1) be a constant and η1 < η0 < 1 be defined as √ 3 3 − (1 + α) 1 η0 = √ . , η1 = √ 2 α +3+α 3 3 + (1 + α)
For every given β > 0, 0 < α < 1, there exists a class of initial data (u0 , v0 ), satisfying the compatibility conditions u0 (0) + v0 (0) = −η[u0 (0) − v0 (0)], u0 (1) + v0 (1) = α[u0 (1) − v0 (1)] − β[u0 (1) − v0 (1)]3 , with finite total variations V[0,1] (u0 ) and V[0,1] (v0 ), such that i. If either 0 < η < η1 or η1−1 < η < ∞, then lim [V[0,1] (u(·, t)) + V[0,1] (v(·, t))] < ∞;
t→∞
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FIG. 1. The rapid change of self-excited energy defined in (1.2) from Example 3. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
ii. If either η1 < η < η0 or η0−1 < η < η1−1 , then lim V[0,1] (u(·, t)) = ∞,
t→∞
lim V[0,1] (v(·, t)) = ∞;
t→∞
iii. There exists a ηc ∈ (η1 , η0 ) such that for any η ∈ [ηc , ηc−1 ] but η = 1, we have V[0,1] (u(·, t)) = O(exp(c1 t)),
V[0,1] (v(·, t)) = O(exp(c2 t)),
for some positive constants c1 and c2 . That is, the growth rates of the total variation of u(·, t) and v(·, t) are exponential as t → ∞. In Lemma 1, conclusions (ii) and (iii) imply that (u, v) may behavior very irregularly even when the initial data (u0 , v0 ) are very regular. The total variation of u(·, t) and v(·, t) being large indicates that u and v undergo rapid fluctuations. Moreover, conclusion (iii) states that the growth rates of the total variation of u(·, t) and v(·, t) may be exponential under certain circumstances, which corresponds to chaotic dynamics of (u, v) [7]. Since (wx , wt ) is topologically conjugate to (u, v), (u, v) is equivalent to (wx , wt ). Hence, chaotic wx and wt will be very sensitive to computation errors, and thus in this case it is challenging for applying traditional high-order finite difference approximations to wx and wt to obtain a satisfactory accuracy. Figure 1 depicts such the variations of total energy E(t) that leads to chaotic dynamics of (wx , wt ). Several finite difference schemes have been used very widely for linear second-order wave equation [25–29] without first transforming it into a system of first-order equations. Such a direct approach often solves fewer dependent variables and circumvents the possible difficulty related to the equivalence, or lack thereof, between the first- and second-order systems. In [30, 31], the authors proposed high-order finite difference approximations for the linear wave equation with discontinuous coefficients, where the global domain is first divided at those discontinuities into subdomains, and then perform discretization on each subdomain using summation by parts operators before patch them together back using a projection method [31] or a penalty method [30]. Numerical Methods for Partial Differential Equations DOI 10.1002/num
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However, existing numerical methods in current literature, including those mentioned above, may not be suitable for solving system (1.1) discussed in this article due to the nonexistence of regular solutions. The inherent feature of (1.1)—the generated waves not necessary being smooth or even continuous—requires us to consider its solutions completely under the framework of weak (generalized) solutions. Our proposed approach, boundary integration method (BIM), renders us to obtain the weak solution with high accuracy and low computational cost. Moreover, if the wave solution is smooth enough, a high-order implicit finite difference scheme is developed with a fast convergence through an integration on the boundary. The “implicit” idea here is to combine the unknowns and their derivatives over neighboring grid points to derive high-order approximations (see, e.g., [32, 33]). Such implicit schemes include combined compact difference schemes, which have been successfully applied to both steady and unsteady convection–diffusion equations [34– 36]. Our proposed scheme extends the technique developed in [37] for solving heat equations in the time direction to the spatial variable for solving (1.1) under our theoretical development. To the best of our knowledge, effective numerical method for computing the weak solution of (1.1) has not been seen in current literature. The main contributions of this article are given below: 1. We develop a weak solution framework for the first-order system of (1.1) that is not available in current literature and is crucial for the proposed numerical approaches in this article. Inspired by [4, 5], we establish two interval mappings that incorporate the boundary conditions and can characterize the dynamics of the Riemann invariants (1.5) and the explicit iterative representation for (u(x, t), v(x, t)) is obtained for t ≥ 0. 2. Then we show that for a given initial condition (w0 , w1 ) ∈ L∞ ([0, 1]) × L∞ ([0, 1]), the unique weak solution w ∈ L∞ ([0, 1] × [0, ∞)) can be expressed as
t+x
1−x+t
u(0, τ )dτ −
w(x, t) = x
v(1, τ )dτ + w0 (x). 1−x
Here the initial condition (w0 , w1 ) is not required to satisfy the compatible condition. This expression allows us to use advanced integration methods such as adaptive quadrature method to obtain the numerical solution with high accuracy. The proposed approach does not require to solve the equation on spatial grid points, instead, it only needs to compute the integrations on the boundary. Thus, our approach not only significantly reduces the regularity requirement for the solution but also has low computational complexity. 3. Moreover, our proposed approach allows us to obtain the numerical solution near the final time T that depends on little prior inner grid solution. Hence the proposed approach is quite efficient when the computation of long-term dynamics is required. 4. Furthermore, if compatibility conditions hold with a sufficient degree, then based on the representation of u and v that include the underlying boundary conditions, we develop a sixth-order finite-difference scheme (5.32) Aw(t) = Bw (t) + w(0, t)e1 , where A and B are constant sparse matrices independent of the time variable t, and w (t) is available by two interval mappings that are determined by initial condition (w0 , w1 ). Our proposed scheme has its own adaptive regularity, i.e., if at the time instance t, the wave solution w(·, t) ∈ C k ([0, 1]), then the corresponding finite difference solution has k-order convergence at t, where 1 ≤ k ≤ 6. Numerical Methods for Partial Differential Equations DOI 10.1002/num
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The article is organized as follows. In Section 2, we provide a general framework for solving the canonical first-order hyperbolic system associated with wave reflections on the boundary. The weak (or generalized) solution is shown to be characterized by two interval mappings that govern the traveling wave and its interaction with the boundary. Based on the results obtained in Section 2, we construct two interval mappings for the Riemann invariant of (1.1) in Section 3. Thus, explicit iterative expression for (wx , wt ) is established in this section. In Section 4, we show that the weak solution of (1.1) can be obtained by the integration of the corresponding Riemann invariant on the boundary and the Gauss–Lobatto integration is proposed for the approximation. Section 5 develops a sixth-order computational scheme for solving (1.1). Several numerical examples are provided in Section 6 to demonstrate the effectiveness of the proposed approaches. The paper ends with concluding remarks in Section 7. II. FIRST-ORDER HYPERBOLIC SYSTEMS
In this section, we consider the canonical first-order hyperbolic system
∂ u(x, t) ∂ u(x, t) 1 0 , 0 < x < 1, t > 0, = 0 −1 ∂x v(x, t) ∂t v(x, t)
(2.6)
with “reflected” boundary conditions v(0, t) = ϕ(u(0, t)),
u(1, t) = φ(v(1, t)), t > 0,
(2.7)
where both φ and ϕ are continuous functions on IR. The boundary conditions (2.7) result in wave reflection on the boundary whose dynamics can be characterized by incoming/outgoing characteristic variables. By incoming characteristic, we mean a characteristic that enters the domain at the boundary; an outgoing characteristic is one that leaves the domain (see Fig. 2). Definition 1. A vector function (u, v) : [0, T ] → L∞ ([0, 1]) × L∞ ([0, 1]) is a weak solution of (2.6) for (u(·, 0), v(·, 0)) = (u0 , v0 ) ∈ L∞ ([0, 1]) × L∞ ([0, 1]) if 1. for any function h ∈ C0∞ ([0, 1] × (0, T ]) (i.e., h is C ∞ function on [0, 1] × (0, T ] with h(0, t) = h(1, t) = 0 for 0 < t ≤ T and h(x, T ) = 0 for 0 ≤ x ≤ 1), we have 1 T 1 1 T 1 (uht − uhx )dxdt = h|t=0 u0 dx, (vht + vhx )dxdt = h|t=0 v0 dx 0
0
0
and 2. v(0, t) = ϕ(u(0, t)),
0
0
0
u(1, t) = φ(v(1, t)) hold for all t > 0.
Theorem 1. The weak solution of the hyperbolic system (2.6) is unique. In other words, (2.6) is well-posed. Proof. Suppose that (u1 , v1 ) and (u2 , v2 ) are two weak solutions of (2.6). Let u˜ = u1 −u2 and ˜ 0) = v(x, ˜ 0) = 0 and for any function h ∈ C0∞ ([0, 1] × [0, T ]) v˜ = v1 − v2 . Then we have u(x, we have T 1 T 1 (uh ˜ t − uh ˜ x )dxdt = 0, (vh ˜ t + vh ˜ x )dxdt = 0. 0
0
0
0
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Classical result yields that u˜ is a constant along the characteristic x + t = constant and v˜ remains unchanged along the characteristic x − t = constant. We first assume T ≤ 1. For x + t = c with 0 ≤ c ≤ 1, we have u(x, ˜ t) = u(x ˜ + t, 0) = u˜ 0 (x + t) = 0, and thus u˜ ≡ 0 in 0 ≤ x + t ≤ 1. In the region, 1 − x ≤ t ≤ T , along the characteristic x + t = c with 1 ≤ c ≤ 2, we have u(x, ˜ t) = u(1, ˜ c − 1) = u1 (1, c − 1) − u2 (1, c − 1) = φ(v1 (1, c − 1)) − φ(v2 (1, c − 1)) = φ(v1 (c, 0)) − φ(v2 (c, 0)) = φ(v0 (c)) − φ(v0 (c)) = 0. Hence we have u1 (x, t) = u2 (x, t) for 0 ≤ t ≤ T . Similarly, for x − t = c with 0 ≤ c ≤ 1, one can see v(x, ˜ t) = v(x ˜ − t, 0) = v˜0 (x − t) = 0, which gives v(x, ˜ t) ≡ 0 in 0 ≤ x − t ≤ 1. In the region, x ≤ t ≤ T , along t − x = c with 0 ≤ c ≤ 1, we have v(x, ˜ t) = v(0, ˜ c) = v1 (0, c) − v2 (0, c) = ϕ(u1 (0, c)) − ϕ(u2 (0, c)) = ϕ(u1 (c, 0)) − ϕ(u2 (c, 0)) = ϕ(u0 (c)) − ϕ(u0 (c)) = 0. Thus, we obtain v1 (x, t) = v2 (x, t) for 0 ≤ t ≤ T . When T ≥ 1, the conclusion now can follow by mathematical induction. We next show that under above setting, the hyperbolic system (2.6) will admit a (weak) solution. According to classical theory, the general solution of (2.6) is of the form u(x, t) = f (x + t),
v(x, t) = g(x − t),
where f and g are arbitrary functions defined on IR. The characteristics x ± t of (2.6) are the quantities that actually propagate in the flow. The general solution indicates the wave traveling to the left at a unit speed is characterized by u, while the wave moving to the right at the same speed is determined by v. Waves interact at the boundary x = 0, x = 1 through functions ϕ and φ. Since the wave speed is one, it is not difficult to see that if a wave reach to the boundary it will hit the boundary again at t = 2 with the same moving direction. To see that, by noting u(x, t) = f (x +t), we immediately have u(1, 0) = f (1) = u(0, 1) and thus the wave of u starting at x = 1 arrives at the left boundary x = 0 at t = 1. Then it is transformed to the wave of v by v(0, 1) = ϕ(u(0, 1)) at x = 0. Since v(x, t) = g(x − t) = v(1, 1 − x + t), we have v(1, 2) = v(0, 1) = ϕ(u(0, 1)). Based on the wave reflection, we next construct the solution (u, v) of (2.6). If 0 ≤ t ≤ 1 − x, then we have u(x, t) = u(x + t, 0) = u0 (x + t). When 1 − x ≤ t ≤ 2 − x, by noting that u(x, t) = f (x + t) = u(1, x + t − 1) Numerical Methods for Partial Differential Equations DOI 10.1002/num
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and the boundary condition u(1, t) = φ(v(1, t)), one has u(x, t) = φ(v(1, x + t − 1)) = φ(v(2 − x − t, 0) = φ(v0 (2 − x − t)). Further if 2 − x ≤ t ≤ 2, then we have u(x, t) = f (x + t) = u(1, x + t − 1) = φ(v(1, x + t − 1)) = φ(v(0, 2 − x − t)) = φ(ϕ(u(0, x + t − 2))) = φ(ϕ(u0 (x + t − 2))), which implies that for x ∈ [0, 1] wave of u goes back to the same location x with the same moving direction at the time t = 2. In other words, the wave has a “period of two” in terms of its dynamics. In summary, we have obtained ⎧ ⎪ t ≤ 1 − x, ⎨u0 (x + t), u(x, t) = φ(v0 (2 − x − t)), (2.8) 1 − x < t ≤ 2 − x, ⎪ ⎩ (φ ◦ ϕ)(u0 (x + t − 2)), 2 − x < t ≤ 2. Similarly, the wave of v can be discussed in the same way. When 0 ≤ t ≤ x, it is obvious that v(x, t) = g(x − t) = v(x − t, 0) = v0 (x − t). Next when x ≤ t ≤ x + 1, boundary condition v(0, t) = ϕ(u(0, t)) yields v(x, t) = g(x − t) = v(0, t − x) = ϕ(u(0, t − x)) = ϕ(u0 (t − x)). When x + 1 ≤ t ≤ 2, both boundary conditions v(0, t) = ϕ(u(0, t)) and u(1, t) = φ(v(1, t)) imply v(x, t) = g(x − t) = v(0, −x + t) = ϕ(u(0, −x + t)) = ϕ(u(1, −1 − x + t)) = φ(ϕ(v(1, −1 − x + t)) = φ(ϕ(v0 (2 + x − t))) which indicates the wave starting at x will return to x with the same moving direction at the time t = 2. Therefore, the solution of v on time interval 0 ≤ t ≤ 2 can be expressed as ⎧ ⎪ t ≤ x, ⎨v0 (x − t)), v(x, t) = ϕ(u0 (t − x)), (2.9) x < t ≤ 1 + x, ⎪ ⎩ (ϕ ◦ φ)(v0 (x − t + 2)), 1 + x < t ≤ 2. Figure 2 shows how the wave propagates along the characteristics. More general, we show the existence of the weak solution of (2.6). Theorem 2. Suppose u0 , v0 ∈ L∞ ([0, 1]). For any t > 0, we denote t = 2n + τ where n is a nonnegative integer and 0 ≤ τ < 2. Then the weak solution of (2.6) is given by ⎧ n ⎪ τ ≤ 1 − x, ⎨(φ ◦ ϕ) (u0 (x + τ )), u(x, t) = (φ ◦ ϕ)n (φ(v0 (2 − x − τ ))), (2.10) 1 − x < τ ≤ 2 − x, ⎪ ⎩ n (φ ◦ ϕ) (φ ◦ ϕ(u0 (x + τ − 2))), 2 − x < τ ≤ 2, Numerical Methods for Partial Differential Equations DOI 10.1002/num
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FIG. 2. The wave propagating path of u(x, t) and v(x, t) along characteristics. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
and
⎧ n ⎪ τ ≤ x, ⎨(ϕ ◦ φ) (v0 (x − τ )), n v(x, t) = ϕ ◦ φ) (ϕ(u0 (τ − x))), x < τ ≤ 1 + x, ⎪ ⎩ n (ϕ ◦ φ) (ϕ ◦ φ(v0 (x − τ + 2))), 1 + x < τ ≤ 2,
(2.11)
where (ϕ ◦ φ)n represents the n-times iterative composition of ϕ ◦ φ with n = 0, 1, 2, . . . and (ϕ ◦ φ)0 = identity. Proof. We have already shown that when n = 0 the claim is true. The rest for showing (2.8) and (2.9) follows directly by induction. The following corollary states the regularity of the solution of (2.6) whose proof is straightforward. Corollary 1.
Suppose u0 , v0 ∈ C k ([0, 1]) that satisfy the following compatible conditions j dj ) j d v0(j ) (0) = (−1)j j ϕ(u0 (x)) , u(j (1) = (−1) (x)) (2.12) φ(v 0 0 dx dx x=0 x=1
for j = 0, 1, 2, . . . , k. Then the solution of (2.6) is of class C k . III. ITERATIONS OF THE RIEMANN INVARIANT OF (1.1)
Based on results presented in previous section, now we are ready to discuss the system (1.1). By Riemann invariants (1.5), or equivalently, let wx = u + v, wt = u − v, then the Eq. (1.1) can be converted into a first-order hyperbolic system in the form of
∂ u(x, t) ∂ u(x, t) 1 0 = , 0 < x < 1, t > 0. (3.13) 0 −1 ∂x v(x, t) ∂t v(x, t) Numerical Methods for Partial Differential Equations DOI 10.1002/num
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The boundary condition at the left end x = 0 now becomes u(0, t) + v(0, t) = −η[u(0, t) − v(0, t)], or v(0, t) =
η+1 u(0, t), t > 0, η−1
(3.14)
and hence we have ϕ(ξ ) =
η+1 ξ. η−1
While at the right-handed side (x = 1) the boundary condition satisfies β[u(1, t) − v(1, t)]3 + (1 − α)[u(1, t) − v(1, t)] + 2v(1, t) = 0. If β = 0, then we have u(1, t) =
α+1 v(1, t). α−1
(3.15)
If β > 0, notice that when 0 < α < 1 for each given x ∈ R the real function y = p(x) is uniquely defined through the following implicit cubic equation βy 3 + (1 − α)y + 2x = 0, by the Cardano’s formula
y=
3
x √ − + D+ β
3
−
x √ − D, β
where D=
x2 (1 − α)3 + > 0. 27β 3 β2
Hence we can write u(1, t) = v(1, t) + p(v(1, t)),
t > 0.
(3.16)
Thus we obtain φ(ξ ) = ξ + p(ξ ) if β = 0;
or
φ(ξ ) =
α+1 ξ if β = 0. α−1
The initial conditions are now in the form of u(x, 0) = u0 (x) ≡
w0 (x) + w1 (x) , 2
v(x, 0) = v0 (x) ≡
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w0 (x) − w1 (x) . 2
(3.17)
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According to Theorem 2, we have the following: Theorem 3.
and
The Riemann invariants of (1.1) can be expressed in an iterative form as ⎧ n ⎪ τ ≤ 1 − x, ⎨(φ ◦ ϕ) (u0 (x + τ )), n u(x, t) = (φ ◦ ϕ) (φ(v0 (2 − x − τ ))), 1 − x < τ ≤ 2 − x, ⎪ ⎩ n+1 (φ ◦ ϕ) (u0 (x + τ − 2)), 2 − x < τ ≤ 2, ⎧ n ⎪ τ ≤ x, ⎨(ϕ ◦ φ) (v0 (x − τ )), n v(x, t) = (ϕ ◦ φ) (ϕ(u0 (τ − x))), x < τ ≤ 1 + x, ⎪ ⎩ (ϕ ◦ φ)n+1 (v0 (x − τ + 2)), 1 + x < τ ≤ 2,
(3.18)
(3.19)
where t = 2n + τ with 0 ≤ τ < 2 and (ϕ ◦ φ)n represents the n-times iterative composition of ϕ ◦ φ. Using (3.18) and (3.19), u and v of the PDE (3.13) can be represented by the iterations of the maps ϕ ◦ φ and φ ◦ ϕ on R, respectively. The dynamical behavior of u and v can be completely characterized by these two one-dimensional maps. IV. NUMERICAL INTEGRATION ON THE BOUNDARY
From previous section, we have obtained the explicit expression of the Riemann invariants (u, v) of (1.1) and thus (wx , wt ) can be computed by wx (x, t) = u(x, t) + v(x, t) and
wt (x, t) = u(x, t) − v(x, t),
where u and v are given by (3.18) and (3.19). Since w(x, 0) = w0 (x) and wt (x, 0) = w1 (x) are known, we formally have t [u(x, t) − v(x, t)]dt + w0 (x). w(x, t) = 0
Thus solving (1.1) turns to be the study of above integration. The next theorem shows that w(x, t) can be obtained by the integration of the Riemann invariants u only at the boundary x = 0 and v at x = 1, respectively. Theorem 4. For given w0 , w1 ∈ L∞ ([0, 1]), the weak solution w ∈ L∞ ([0, 1] × [0, T ]) of (1.1) can be expressed as x+t 1−x+t w(x, t) = u(0, τ )dτ − v(1, τ )dτ + w0 (x). (4.20) x
1−x
Moreover, if w0 ∈ C 1 ([0, 1]), w1 ∈ C([0, 1]) and the following compatible condition w0 (0) = −ηw1 (0),
w0 (1) = αw1 (1) − βw13 (1)
is satisfied, then the solution w ∈ C 1 ([0, 1] × [0, T ]). Numerical Methods for Partial Differential Equations DOI 10.1002/num
(4.21)
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In general, if the initial condition (w0 , w1 ) ∈ C k ([0, 1]) × C k−1 ([0, 1]) and satisfies the compatible condition (2.12) for j = 1, 2, . . . , k, then the solution w ∈ C k ([0, 1] × [0, T ]). Proof. Let 0 ≤ x ≤ 1 and t > 0. Notice that u is constant along its characteristic x + t = constant, we have u(x, t) = u(0, x + t). Thus
t
t
u(x, τ )dτ =
x+t
u(0, x + τ )dτ =
0
0
u(0, τ )dτ . x
Similarly, v remains constant along its characteristic t − x = constant and hence v(x, t) = v(1, 1 − x + t). This yields
t
t
v(x, τ )dτ =
1−x+t
v(1, 1 − x + τ )dτ =
0
v(1, τ )dτ .
0
1−x
Since w0 , w1 ∈ L∞ ([0, 1]) and ϕ, φ are continuous on IR, its Riemann invariants u, v ∈ L∞ ([0, 1] × [0, T ]). Therefore, the solution
t
w(x, t) =
(u(x, t) − v(x, t))dt + w0 (x) 0
x+t
=
1−x+t
u(0, τ )dτ −
v(1, τ )dτ + w0 (x) 1−x
x
is a function belonging to L∞ ([0, 1] × [0, T ]). The rest of the proof is straightforward. Remark 1.
We here make some remarks about Theorem 4:
1. To obtain the weak solution of (1.1), according to (4.20), one only needs to compute the integrations of u(0, τ ) and v(1, τ ) with respect to the time variable τ . This significantly reduces the computation complexity. 2. Suppose that the initial condition (w0 , w1 ) does not satisfy the compatible conditions. If (w0 , w1 ) is smooth, then both u(0, τ ) and v(1, τ ) are continuous in (n, n + 1) but are discontinuous at n = 0, 1, 2, . . . . Let t = n + s with s ∈ [0, 1). Then the integration of u and v can be written, respectively, as
x+t
1
u(0, τ )dτ =
+
x
n+s+x
2
+··· + 1
x
u(0, τ )dτ ,
f or
s+x ≤1
u(0, τ )dτ ,
f or
s+x >1
n
or
x+t
u(0, τ )dt = x
1
2
+
+··· + 1
x
n+s+x n+1
and
1−x+t
1−x
1
v(1, τ )dτ =
+ 1−x
1−x+t
2
+··· + 1
v(1, τ )dτ , n
Numerical Methods for Partial Differential Equations DOI 10.1002/num
f or
s≤x
VAN DER POL TYPE NONLINEAR BOUNDARY CONDITIONS
385
or
1−x+t
v(1, τ )dτ = 1−x
1
+ 1−x
2
1−x+t
+··· + 1
v(1, τ )dτ ,
f or
s > x.
n+1
The above decomposition allows us to use methods of high-order integrations on subintervals where either u(0, τ ) or v(1, τ ) is smooth. Since u and v may have large functional variations, some portions require more attention than others. Thus, to obtain better accuracy and efficiency for each subintegral, it is necessary to use the adaptive quadrature method [38–40] in which the step size of t can be adjusted to be smaller over portions of the curve u and v where a larger functional variation occurs. ˆ = u(0, t) or v(1, t) and consider the integration For convenience of description, we denote w(t)
b
w(t)dt. ˆ a
b The basic idea behind adaptive quadrature is to integrate a w(t)dt ˆ first by using two different numerical integration methods, thus obtaining the approximations I1 and I2 . Suppose I1 is more accurate. Then one uses the relative difference between I1 and I2 as an error estimate. If it is less than a given tolerance ε, accept I1 as the answer. Otherwise, divide the interval [0, T ] into two subintervals, T b 1 w(t)dt ˆ = w(t)dt ˆ + w(t)dt, ˆ = (b + a) 2 0 a and compute the two integrals independently. For each one, compute I1 and I2 , estimate the error, and continue subdividing if necessary. Dividing any given subinterval stops when its contribution to ε is sufficiently small. For instance, one may directly use MATLAB’s build in function integral to perform such adaptive quadratures when applicable. To recover the unknowns w(xi , T ) on all uniform spatial grid points, in the first and last subintervals of the temporal integrations, we also compositely use the following high-order 7-point Gauss–Kronrod rule [38]
d 77[w(a) ˆ + w(b)] ˆ + 432[w( ˆ − f · d) + w( ˆ + f · d)] 1470 a +625[w( ˆ − g · d) + w( ˆ + g · d)] + 672w( ) ˆ , b
w(t)dt ˆ ≈
(4.22)
where 1 d = (b − a), 2
1 = (a + b), 2
√
2 f = √ , 3
1 g=√ . 5
This approach makes possible for us to obtain high-order accuracy without compatible assumptions provided that u0 and v0 are smooth enough. Our approach also has advantage for the computation of long-term dynamics of (1.1). To see that, for 0 ≤ t ≤ 1, it is not difficult to see t t w(0, t) = [u(0, τ ) − v(0, τ )]dτ + w0 (0) = [u0 (τ ) − ϕ(u0 (τ ))] dτ + w0 (0) 0
0
Numerical Methods for Partial Differential Equations DOI 10.1002/num
386
LIU ET AL.
t
t
[u(1, τ ) − v(1, τ )]dτ + w0 (1) =
w(1, t) =
[φ(v0 (1 − t)) − v0 (1 − t)] dτ + w0 (1). 0
0
Then according to the property of the characteristic quadrilateral of wave equation, for 0 ≤ x ≤ 1, we have w(x, 1) = w(0, 1 − x) + w(1, x) − w0 (1 − x), which implies that to compute the solution w at t = 1, we only need to know the solutions on the boundary x = 0, 1 with 0 ≤ t ≤ 1. Similarly, when t = 2, we have w(x, 2) = w(0, 2 − x) + w(1, 1 + x) − w(1 − x, 1). Note that once the values of w(0, t) and w(1, t) are known for 0 ≤ t ≤ 2, there is no more integration required. Thus, suppose that we would like to compute the long-term behavior of (1.1). Let t = n + s where n > 0 is a positive integer and 0 ≤ s < 1. Further, denote the region = {(x, t) : 0 ≤ x ≤ 1, n ≤ t ≤ n + 1}. Then we only need to compute w(0, t) and w(1, t) for 0 < t ≤ n to obtain w(x, n), and the solution on is given by
t
[u(x, τ ) − v(x, τ )]dτ + w(x, n)
w(x, t) =
n
x+s
=
1−x+s
u(0, n + τ )dτ −
v(1, n + τ )dτ + w(x, n). 1−x
x
The significant point here is that to compute the solution on for any n > 0, we need not know the solution on 0 < x < 1, j − 1 < t < j , j = 1, . . . , n. This greatly reduces the computational complexity if we are only interested in the dynamics near the final time n ≤ t ≤ n + s.
V. HIGH-ORDER FINITE DIFFERENCE APPROACH
In this section, based on the obtained results in previous section, we present a new finite difference scheme which can deliver sixth-order accuracy in spatial variable provided that the compatible conditions (2.12) hold for k ≥ 6, and sixth-order numerical integration in time variable by the mentioned approach in last section. More specifically, we first apply Theorem 3 to obtain both u and v, and then we have wx = u + v,
wt = u − v,
which are now available for following discussion. Let the spatial mesh grid points be x0 = 0,
x1 = hx , . . . , xj = j hx , . . . , xn = nhx = 1.
For a fixed time t, we denote wj = w(xj , t) and wj(k) =
∂k w(x, t)|x=xj ∂x k
for k = 1, 2, · · · .
Numerical Methods for Partial Differential Equations DOI 10.1002/num
VAN DER POL TYPE NONLINEAR BOUNDARY CONDITIONS
387
By using Taylor’s expansions with some manipulations (see Appendix of [37]), for any 3 ≤ j ≤ n − 2, we have 1.5wj +2 − 15wj +1 − 10wj + 30wj −1 − 7.5wj −2 + wj −3 = −30hx wj + O(h6x ),
(5.23)
−wj +2 + 7.5wj +1 − 30wj + 10wj −1 + 15wj −2 − 1.5wj −3 = 30hx wj −1 + O(h6x ).
(5.24)
and
The sum of (5.23) and (5.24) yields hx 1 (wj −3 − 15wj −2 − 80wj −1 + 80wj + 15wj +1 − wj +2 ) = (wj −1 + wj ) + O(h6x ) 120 2 (5.25) for j = 3, 4, . . . , n − 2. Next we establish two more difference equations near the boundary x = 0. Again we apply Taylor’s expansions of w at w0 : 1 1 1 1 5 (5) w1 = w0 + hx w0 + h2x w0 + h3x w0 + h4x w0(4) + h w + O(h6x ) 2 6 24 120 x 0 4 2 4 w2 = w0 + 2hx w0 + 2h2x w0 + h3x w0 + h4x w0(4) + h5x w0(5) + O(h6x ) 3 3 15 9 9 27 81 w3 = w0 + 3hx w0 + h2x w0 + h3x w0 + h4x w0(4) + h5x w0(5) + O(h6x ) 2 2 8 40 32 32 128 w4 = w0 + 4hx w0 + 8h2x w0 + h3x w0 + h4x w0(4) + h5 w (5) + O(h6x ) 3 3 15 x 0 25 125 3 625 4 (4) 625 5 (5) w5 = w0 + 5hx w0 + h2x w0 + h w + h w + h w + O(h6x ) 2 6 x 0 24 x 0 24 x 0 which gives −
50 25 137 w0 + 25w1 − 25w2 + w3 − w4 + w5 = 5hx w0 + O(h6x ). 12 3 4
If we expand w at w1 , then we have 1 1 1 1 5 (5) h w + O(h6x ) w0 = w1 − hx w1 + h2x w1 − h3x w1 + h4x w1(4) − 2 6 24 120 x 1 1 1 1 1 5 (5) w2 = w1 + hx w1 + h2x w1 + h3x w1 + h4x w1(4) + h w + O(h6x ) 2 6 24 120 x 1 4 2 4 w3 = w1 + 2hx w1 + 2h2x w1 + h3x w1 + h4x w1(4) + h5x w1(5) + O(h6x ) 3 3 15 9 9 27 81 w4 = w1 + 3hx w1 + h2x w1 + h3x w1 + h4x w1(4) + h5x w1(5) + O(h6x ) 2 2 8 40 32 32 128 w5 = w1 + 4hx w1 + 8h2x w1 + h3x w1 + h4x w1(4) + h5 w (5) + O(h6x ) 3 3 15 x 1 Numerical Methods for Partial Differential Equations DOI 10.1002/num
(5.26)
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LIU ET AL.
that gives 325 25 5 w1 + 50w2 − 25w3 + w4 − w5 = 25hx w1 + O(h6x ). 12 3 4
−5w0 −
(5.27)
By adding (5.26) and (5.27) and then dividing both sides by 30, one can get −
197 25 300 100 25 3 w0 − w1 + w2 − w3 + w4 − w5 = 360 360 360 360 360 360 hx (w + 5w1 ) + O(h6x ) 6 0
(5.28)
By using the same technique, one can have 1 hx (−6w0 − 125w1 + 80w2 + 60w3 − 10w4 + w5 ) = (w1 + 2w2 ) + O(h6x ) 180 3
(5.29)
Similarly, we establish another two equations near the boundary x = 1 as 1 (−wn−5 + 10wn−4 − 60wn−3 − 80wn−2 + 125wn−1 + 6wn ) = 180 hx (2wn−2 + wn−1 ) + O(h6x ) 3
(5.30)
and 1 (3wn−5 − 25wn−4 + 100wn−3 − 300wn−2 + 25wn−1 + 197wn ) = 360 hx + wn ) + O(h6x ). (5wn−1 6
(5.31)
In summary, we have obtained the following approximated difference equations with sixth-order accuracy: 1 hx (wj −3 − 15wj −2 − 80wj −1 + 80wj + 15wj +1 − wj +2 ) = (wj −1 + wj ) 120 2 for j = 3, 4, . . . , n − 2, and four equations near the boundaries 1 hx (−197w0 − 25w1 + 300w2 − 100w3 + 25w4 − 3w5 ) = (w0 + 5w1 ) 360 6 1 hx (−6w0 − 125w1 + 80w2 + 60w3 − 10w4 + w5 ) = (w1 + 2w2 ) 180 3 1 hx (−wn−5 + 10wn−4 − 60wn−3 − 80wn−2 + 125wn−1 + 6wn ) = (2wn−2 + wn−1 ) 180 3 1 hx + wn ). (3wn−5 − 25wn−4 + 100wn−3 − 300wn−2 + 25wn−1 + 197wn ) = (5wn−1 360 6 Next we compute w(0, t) by integration over the boundary x = 0. By the boundary condition (v(0, t) = η+1 u(0, t)) at x = 0, we have η−1 w(0, t) = 0
t
2 [u(0, τ ) − v(0, τ )]dτ + w0 (0) = 1−η
t
u(0, τ )dτ + w0 (0), 0
Numerical Methods for Partial Differential Equations DOI 10.1002/num
VAN DER POL TYPE NONLINEAR BOUNDARY CONDITIONS
389
which can be computed efficiently via, for example, adaptive quadrature method. Again here u(0, τ ) is known according to (3.18). We remark that the error in above quadrature is the only temporal error in this scheme. Notice that for j = 0, 1, 2, . . . , n we already know ∂ w(x, t) = u(xj , t) + v(xj , t), wj = ∂x x=xj where u(xj , t) and v(xj , t) are computed by (3.18) and (3.19), respectively. Thus wj now is known. For clarity, we denote w (t) = [w0 , w1 , w2 , . . . , wn ] , T
w(t) = [w0 , w1 , w2 , . . . , wn ]T ,
and then gather all above approximations in to the following linear system Aw(t) = Bw (t) + w0 e1 ,
(5.32)
where ⎡
1 ⎢ ⎢ ⎢ ⎢−197/360 ⎢ ⎢ ⎢ ⎢ −6/180 ⎢ ⎢ ⎢ ⎢ 1/120 A=⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎣ 0
0
0
−25/360
300/360
−125/180
0 −100/360
80/180
−15/120 .. . .. . .. . .. . ⎡
−3/360
25/360 −10/180
60/180
···
0
1/180
0
..
.
0
..
.
0
..
.
0
..
.
0
−80/120 .. .
80/120 .. .
15/120 .. .
−1/120 .. .
1/120
−15/120
−80/120
80/120
15/120
−1/120
−1/180
10/180
−60/180
−80/180
125/180
6/180
3/360
−25/360
100/360
−300/360
25/360
197/360
0 ⎢ ⎢1/6 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ B=⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎣ 0
0
0
0
5/6
0
0 .. .
1/3
2/3
0
1 .. . .. . .. . .. .
1 .. . .. . .. . .. .
0 .. . .. . .. . .. .
0 .. . .. . .. . .. . .. . .. . .. .
0 .. . .. . .. . .. .
0 .. . .. . .. . .. .
0 .. . .. . .. . .. .
0
1
1
0 .. .
2/3
1/3
0
5/6
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎤ 0 .. ⎥ .⎥ ⎥ ⎥ .. ⎥ .⎥ ⎥ .. ⎥ ⎥ .⎥ ⎥ ⎥, 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎦ 1/6
and e1 is first column of identity matrix with dimension (n + 1). The nice feature of this approach is that we can obtain the approximated solution on any time instance without requiring any explicit information of prior time solution except at the left boundary x = 0. This sixth-order finite difference scheme (FD-6) turns out to be very efficient since it only needs a boundary integration to approximate w(0, t) and then a linear system solving. This approach minimizes the error accumulation in a significant way. Moreover, the approach remains valid even if the solution Numerical Methods for Partial Differential Equations DOI 10.1002/num
390
LIU ET AL.
w ∈ C k ([0, 1] × [0, T ]) for 1 ≤ k < 6, since in that case we still have Taylor expansion up to k + 1 term, and of course the convergence will be k th-order. In general, its high-order accuracy relies on the regularity of compatibility conditions. The higher regularity of compatibility conditions holds, the higher-order accuracy it will be attained.
VI. NUMERICAL EXAMPLES
In this section, we present numerical tests with three different methods developed in our article. All experiments are implemented using MATLAB 2013a on a Laptop PC with Intel(R) Core(TM) i3-3120M CPU @ 2.50GHz. In discretization, we choose identical spatial and temporal step size h = x = t. The pair (wt , wx ) in our FD-6 and BIM approaches is computed directly by (3.18) and (3.19) with wx (x, t) = u(x, t) + v(x, t),
wx (x, t) = u(x, t) − v(x, t).
When the exact solutions are unknown, we use the approximation obtained by our proposed BIM with the computed finest mesh (e.g. h = 1/10240) as the benchmark reference. Otherwise, we will use the exact solution as the reference. In particular, we measure the maximum absolute error eh of current approximation wh (x, T ) by subtracting it from the reference values over their shared spatial grid points at final time T . As in [41], the experimental order of convergence can be estimated by R = log2
e2h eh
,
which is close to 2 for the second-order scheme (FD-2) and 6 for the sixth-order scheme (FD-6). In particular, the FD-6 sparse linear system can be solved efficiently by MATLAB’s sparse direct backslash solver, and the integration can be obtained by adaptive quadrature function integral. Example 1.
We first consider a linear case (β = 0) with initial conditions w0 (x) = − cos(κx)/κ
and
w1 (x) = sin(κx),
which does not satisfy the compatible condition even for k = 0. Thus, the solution w is discontinuous at t = n but it is smooth on (n, n + 1) for n = 0, 1, · · · . Let u0 (x) = sin(κx)
and
v0 (x) = 0.
It follows from β = 0 that φ ◦ ϕ = ϕ ◦ φ is just a scalar multiplication. We denote the corre. Notice that with α = 1/2, η = −2 we would get γ = −1. sponding scalar by γ := (α+1)(η+1) (α−1)(η−1) By expressions (3.18) and (3.19), for any t = 2n + r with 0 ≤ r ≤ 2, we have (notice we used v0 (x) = 0) ⎧ n ⎪ r ≤ 1 − x, ⎨γ sin(κ(x + r)), u(x, t) = 0, 1 − x < r ≤ 2 − x, ⎪ ⎩ n+1 γ sin(κ(x + r − 2)), 2 − x < r ≤ 2, Numerical Methods for Partial Differential Equations DOI 10.1002/num
(6.33)
VAN DER POL TYPE NONLINEAR BOUNDARY CONDITIONS TABLE I.
391
Errors for solving Example 1 at T = 6 (κ = 70, β = 0, α = 1/2, η = −2). FD-2
Our FD-6
Our BIM
1/h
Error
R
CPU
Error
R
CPU
Error
R
CPU
40 80 160 320 640 1280 2560 5120
2.9e-02 1.2e-02 4.1e-03 1.4e-03 5.7e-04 2.4e-04 1.1e-04 5.3e-05
– 1.3 1.6 1.5 1.4 1.2 1.1 1.1
0.007 0.013 0.028 0.060 0.148 0.369 1.091 4.119
3.5e-03 5.8e-05 5.0e-07 4.9e-09 6.3e-11 9.3e-13 1.6e-14 4.4e-15
– 5.9 6.9 6.7 6.3 6.1 5.8 1.9
0.009 0.009 0.010 0.010 0.012 0.012 0.014 0.028
1.3e-12 2.7e-15 1.1e-15 1.1e-15 1.2e-15 1.2e-15 1.2e-15 1.0e-15
– 8.9 1.2 – – – – –
0.017 0.016 0.017 0.017 0.021 0.019 0.022 0.032
TABLE II.
Errors for solving Example 1 at T = 7 (κ = 7, β = 0, α = 1/2, η = −2). FD-2
Our FD-6
Our BIM
1/h
Error
R
CPU
Error
R
CPU
Error
R
CPU
160 320 640 1280 2560 5120
5.9e-04 3.2e-04 1.7e-04 8.4e-05 4.2e-05 2.1e-05
– 0.9 0.9 1.0 1.0 1.0
0.068 0.146 0.188 0.757 1.432 4.537
1.3e-03 6.7e-04 3.4e-04 1.7e-04 8.4e-05 4.2e-05
– 1.0 1.0 1.0 1.0 1.0
0.014 0.010 0.011 0.018 0.014 0.044
2.5e-16 2.8e-16 5.6e-16 2.9e-16 7.1e-16 5.7e-16
– – – – – –
0.016 0.019 0.013 0.016 0.021 0.132
and
v(x, t) =
⎧ ⎪ ⎨0,
η+1 n γ ⎪ η−1
⎩ 0,
r ≤ x, sin(κ(r − x)), 1 + x < r ≤ 2.
x < r ≤ 1 + x,
(6.34)
With these expressions, the exact solution is given by the formula in Theorem 4. In Tables I and II, we report the error results and computation time of three different methods with T = 6 and T = 7, respectively. We compare our proposed methods with a standard secondorder central finite difference method (FD-2) through several numerical examples. It can be seen that violating compatibility condition renders FD-2 method to be only first-order accuracy. While the convergence order of our FD-6 method depends on whether w(x, T ) crosses nonsmooth points in space. The FD-6 method will be only first-order accurate in the presence of nonsmooth spatial points (see Table II), but it shows a sixth-order accuracy otherwise (see Table I). Notice the nonsmooth spatial points are due to the violation of compatibility. Here we used the obtained analytic solution as the reference. The “Error” columns show that our BIM almost achieves a machine-level accuracy, which is attributed to the employed highly accurate adaptive Gauss-Lobatto quadrature rule. Hence there is no need to report the convergence order (denoted by “–”) of BIM since it reaches the machine precision. The error is affected by the default absolute error tolerance (1e-10) and relative error tolerance (1e-6) for the numerical integration function integral. Further mesh refinements in our BIM may lead to slightly oscillations in errors (at the machine precision level) since heavier calculations may introduce extra round-off errors. We should not expect any significant reduction in error by tightening those tolerances. Thus, it is reasonable to use our proposed BIM as reference in the following examples with exact solutions unknown. The “CPU” columns illustrate that our FD-6 method is much faster than the FD-2 method, and our BIM is far more Numerical Methods for Partial Differential Equations DOI 10.1002/num
392
LIU ET AL. TABLE III.
Errors for solving Example 2 with T = 7 (κ = 1, β = 1, α = 1/2, η = 1/2). FD-2
Our FD-6
Our BIM
1/h
Error
R
CPU
Error
R
CPU
Error
R
CPU
160 320 640 1280 2560 5120 10240
1.8e-02 9.8e-03 4.9e-03 2.7e-03 1.5e-03 7.7e-04 4.0e-04
– 0.9 1.0 0.9 0.9 0.9 1.0
0.036 0.075 0.184 0.470 1.390 4.856 17.126
3.4e-03 1.4e-03 7.2e-04 3.6e-04 1.8e-04 9.0e-05 4.5e-05
– 1.3 1.0 1.0 1.0 1.0 1.0
0.019 0.019 0.021 0.024 0.029 0.038 0.056
3.7e-06 9.6e-07 1.6e-08 6.8e-10 6.2e-12 8.0e-15
– 2.0 5.9 4.6 6.8 9.6
0.034 0.035 0.038 0.042 0.056 0.088 0.170
FIG. 3. The different approximations of w(x, T = 7) in Example 2 (h = 1/160). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
accurate than both of them. The fast speed of FD-6 method comes from our developed boundary integration techniques that reduces the computational complexity.
Example 2. In this example, the initial conditions are the same as in Example 1 and the test parameters are β = 1, α = 1/2, η = 1/2. Notice that the presence of nonlinear term β = 0 will complicate the dynamics of the solution. The results in Table III demonstrate the high-order accuracy of our BIM without compatibility conditions. The corresponding approximated solutions w(x, T = 7) are plotted in Fig. 3. Once again, both FD-2 and FD-6 methods attain only first-order accuracy, but our FD-6 method still achieves better accuracy with much less CPU time. Moreover, our BIM has significant advantage over both FD-2 and FD-6 methods in the sense that it can provide much higher accuracy with a very fast speed. In particular, it costs about 18 s for the FD-2 method to deliver four digits accuracy, while our FD-6 method and BIM need roughly 0.03 s, speeding up over 600 times. Numerical Methods for Partial Differential Equations DOI 10.1002/num
VAN DER POL TYPE NONLINEAR BOUNDARY CONDITIONS
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Consider the initial conditions given by the following C 2 -spline functions: ⎧ ⎪ (x − x1 )3 ⎪ ⎪ , x1 ≤ x ≤ x2 , ⎪ ⎪ h3 ⎪ ⎪ ⎪ ⎪ 3(x − x2 )3 3(x − x2 ) 3(x − x2 )2 ⎪ ⎪ − , x2 ≤ x ≤ x3 , 1+ + ⎪ 2 ⎪ h h h3 ⎪ ⎨ 2 3 1 3(x − x4 ) 3(x − x4 ) 3(x − x4 ) u0 (x) = 1− + , x3 ≤ x ≤ x4 , + 2 20 ⎪ h h3 h ⎪ ⎪ 3 ⎪ ⎪ (x5 − x) ⎪ ⎪ , x4 ≤ x ≤ x5 , ⎪ ⎪ h3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0, elsewhere,
Example 3 ([42]).
⎧ ⎪ (x − y1 )3 ⎪ ⎪ , y1 ≤ x ≤ y2 , ⎪ ⎪ h3 ⎪ ⎪ ⎪ ⎪ 3(x − y2 )3 3(x − y2 ) 3(x − y2 )2 ⎪ ⎪ − , y2 ≤ x ≤ y3 , 1 + + ⎪ ⎪ h h2 h3 ⎪ ⎨ 2 3 1 3(x − y4 ) 3(x − y4 ) 3(x − y4 ) v0 (x) = 1− + + , y3 ≤ x ≤ y4 , 2 20 ⎪ h h h3 ⎪ ⎪ 3 ⎪ ⎪ (y − x) ⎪ ⎪ 5 , y4 ≤ x ≤ y5 , ⎪ ⎪ h3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0, elsewhere, , i = 1, 2, 3, 4, 5. Notice that above initial conditions satwhere h = 15 , xi = 5i , yi = i−1 5 isfy the compatibility condition up to first order, which implies the FD-2 method should give a second-order accuracy. The error results for Example 3 with T = 20 are given in Table IV. The FD-2 method fails to give a satisfactory second-order accuracy until the finest mesh. Instead, our proposed BIM and FD-6 method keep its high-order accuracy, which greatly outperforms the FD-2 method. In Fig. 4, we plot the computed approximations w(x, T ) with different methods to emphasize the possible large approximation error of the FD-2 method. It is worthwhile to point out that the FD-2 method produces some noticeable oscillations, which numerically indicates the possible instability of the FD-2 method under nonlinear chaotic dynamics. We remark that for this particular example both wx (x, t) and wt (x, t) undergo chaotic fluctuations as T increases, which dramatically deteriorates the accuracy of traditional finite difference approximations, such as FD-2 method. Our FD-6 method continues to show certain high-order accuracy since it is based on our developed boundary integration approach. Example 4.
In this example, we choose the initial conditions [4] π x , w1 (x) = 0.2 sin(π x), w0 (x) = 0.2 sin 2
which give u0 (x) = 0.1
π 2
cos
π x + sin(π x) , 2
v0 (x) = 0.1
π 2
cos
π x − sin(π x) . 2
The parameters are set as β = 1, α = 1/2, η = 1.52 to test the chaotic dynamics [2]. In this case, the compatible condition (4.21) hold up to only first order. The error results for Example 4 Numerical Methods for Partial Differential Equations DOI 10.1002/num
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LIU ET AL. TABLE IV.
Errors for solving Example 3 with T = 20 (β = 1, α = 1/2, η = 0.55). FD-2
Our FD-6
Our BIM
1/h
Error
R
CPU
Error
R
CPU
Error
R
CPU
160 320 640 1280 2560 5120 10240
1.1e-01 1.5e-01 1.5e-01 1.7e-01 1.5e-01 1.2e-01 6.4e-02
– −0.4 −0.0 −0.1 0.2 0.3 0.9
0.098 0.209 0.524 1.241 3.824 13.061 40.744
4.6e-02 2.7e-02 4.0e-03 9.7e-04 4.6e-04 2.8e-05 3.0e-06
– 0.8 2.8 2.0 1.1 4.0 3.2
0.386 0.389 0.387 0.412 0.413 0.550 0.425
1.7e-03 6.9e-04 8.4e-05 2.3e-06 2.3e-07 4.4e-10
– 1.3 3.0 5.2 3.3 9.1
0.683 0.839 0.685 0.776 0.846 0.829 1.043
FIG. 4. The approximations of w(x, T = 20) in Example 3 (h = 1/160). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
with T = 7 are given in Table V. In Fig. 5, we plot the computed approximations w(x, T ) with different methods to show the chaotic dynamics. Our FD-6 method and BIM show much faster speed as well as smaller error compared with the FD-2 method. Example 5.
In this example, we choose the initial conditions w0 (x) = 10x 6 (x − 1)6
and
w1 (x) = 10(ex
4 (1−x)4
− 1)
with w0 and w1 satisfying the compatibility conditions up to the sixth order. The error results for Example 4 with T = 3 are given in Table VI. Numerical results indicate that we successfully achieved the anticipated second-order and sixth order accuracy for our FD-2 and FD-6 scheme, respectively. We remark that the slightly increasing errors near the machine-level accuracy for FD-6 scheme with very fine meshes is reasonable, considering the introduced round-off errors as well as condition number of solving the corresponding linear system (5.32). While both our FD-6 scheme and BIM are superior than the FD-2 scheme in both accuracy and efficiency, the Numerical Methods for Partial Differential Equations DOI 10.1002/num
VAN DER POL TYPE NONLINEAR BOUNDARY CONDITIONS TABLE V.
395
Errors for solving Example 4 with T = 7 (β = 1, α = 1/2, η = 1.52). FD-2
Our FD-6
Our BIM
1/h
Error
R
CPU
Error
R
CPU
Error
R
CPU
160 320 640 1280 2560 5120 10240 20480
8.5e-03 1.9e-03 8.4e-04 5.3e-04 3.4e-04 2.0e-04 1.1e-04 5.6e-05
– 2.1 1.2 0.7 0.6 0.8 0.9 0.9
0.034 0.073 0.172 0.441 1.267 4.104 16.048 61.976
1.2e-02 5.1e-03 2.0e-03 8.0e-04 3.5e-04 1.6e-04 8.2e-05 4.1e-05
– 1.2 1.4 1.3 1.2 1.1 1.0 1.0
0.019 0.019 0.021 0.023 0.027 0.037 0.052 0.097
3.3e-05 2.0e-06 1.1e-06 1.2e-08 1.9e-09 1.3e-11 1.8e-14
– 4.1 0.8 6.6 2.6 7.2 9.4
0.037 0.039 0.042 0.049 0.067 0.102 0.172 0.277
FIG. 5. The approximations of w(x, T = 7) in Example 4(h = 1/160). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
BIM shows better robustness since it does not rely on solving any linear systems like (5.32). The corresponding approximated solutions w(x, T = 3) with different methods are plotted in Fig. 6.
VII. CONCLUDING REMARKS
In this article, we provide numerical methods for solving the wave equation with van der Pol type boundary condition. Due to the complexity of the system dynamics, classical approaches are no longer to be satisfactory in most cases due to either the required compatible condition does not hold or the occurrence of chaos for the solution gradient. Different from classical approaches, we develop numerical schemes that only requires the integrations of Riemann invariants on the boundary and obtain the weak solution with high-order accuracy. Moreover, the proposed approach significantly reduces the required computational cost and numerical evidence shows the approach is very cost efficient with high accuracy, although the dynamics of the system is quite complex. Numerical Methods for Partial Differential Equations DOI 10.1002/num
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LIU ET AL. TABLE VI.
Errors for solving Example 4 with T = 3 (β = 1, α = 1/2, η = 1/2).
FD-2
Our FD-6
Our BIM
1/h
Error
R
CPU
Error
R
CPU
Error
R
CPU
40 80 160 320 640 1280 2560 5120
1.1e-04 2.7e-05 6.7e-06 1.7e-06 4.2e-07 1.1e-07 2.6e-08 6.6e-09
– 2.0 2.0 2.0 2.0 2.0 2.0 2.0
0.004 0.007 0.017 0.072 0.164 0.385 0.939 2.666
1.6e-07 2.8e-09 3.9e-11 5.1e-13 8.4e-15 1.7e-14 3.1e-14 6.9e-14
– 5.8 6.2 6.3 5.9 – – –
0.005 0.005 0.005 0.009 0.011 0.014 0.018 0.028
7.8e-16 1.1e-15 1.1e-15 1.1e-15 1.1e-15 1.1e-15 1.0e-15
– – – – – – –
0.008 0.012 0.015 0.018 0.026 0.040 0.072 0.292
FIG. 6. The approximations of w(x, T = 3) in Example 4(h = 1/160). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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