On Spectral Filtering for Discontinuous Galerkin Methods on Unstructured Triangular Grids A. Meister∗ , S. Ortleb† , Th. Sonar‡
Abstract We adapt the concept of spectral vanishing viscosity to a discontinuous Galerkin method solving hyperbolic conservation laws on triangular grids. In this context, modal filtering applied to the elementwise Dubiner expansion of the numerical solution is newly related to a super viscosity formulation based on the associated Sturm-Liouville operator. This connection allows to specify conditions on the filter strength with respect to time step choice and mesh refinement. The small amount of artificial dissipation introduced to the scheme stabilizes the approximation, but weaker oscillations are still present. The digital total variation filter, which has not been applied to DG methods before, reveals to be capable of removing the remaining oscillations in a postprocessing step. Numerical experiments carried out for a two-dimensional linear advection equation confirm the designed order of accuracy for polynomial degrees up to 8. The influence of global and adaptive spectral filtering is investigated as well. In addition, the spectral and DTV filtering techniques are tested for a nonlinear conservation law with discontinuous solution.
1
Introduction
Any kind of numerical method applied to the simulation of hyperbolic conservation laws must be able to deal with discontinuities occurring in the exact entropy solution. In the case of high order methods some sort of supplementary artificial dissipation has to be used as the numerical solutions suffer from the presence of Gibbs oscillations. In the context of high order finite volume methods weighted essentially non-oscillatory (WENO) reconstructions accomplish this task by constructing a preferably non-oscillatory polynomial interpolation of the cell avarages within a given stencil. We refer to [18, 15] for the WENO methodology in the one-dimensional case and to [9] for reconstructions on two-dimensional unstructured grids. Discontinuous Galerkin (DG) methods may use limiters – borrowed from the finite volume methodology as well – to avoid an oscillatory solution. These limiters are often designed to reduce the polynomial degree N near discontinuities to N = 1 and modify the slope depending on neighbouring cell data as in the case of the Runge-Kutta discontinuous Galerkin (RKDG) methods developed by Cockburn, Shu et al. in a series of papers, see the review [4] and references therein. The authors suggest to employ a modified minmod function to define the slope of the limited solution avoiding an approximation of only first order at smooth extrema. Nevertheless, once the minmod limiter is enforced in a given cell, ∗
Fachbereich Mathematik, Universit¨ at Kassel, Heinrich Plett Str. 40, 34132 Kassel, Germany – e-mail:
[email protected] † Fachbereich Mathematik, Universit¨ at Kassel, Heinrich Plett Str. 40, 34132 Kassel, Germany – e-mail:
[email protected] ‡ Institut Computational Mathematics, Technische Universit¨ at Braunschweig, Pockelsstraße 14, 38106 Braunschweig, Germany – e-mail:
[email protected]
1
the information inherent in the higher order coefficients is lost. Consequently, limiters starting from the higher order coefficients and modifying them only when it is needed instead of setting them directly to zero have been suggested in [1, 17] but heavily rely on cartesian structured grids. WENO or HWENO reconstructions, where the costly reconstruction procedure is restricted to regions marked by a shock sensor, have also been applied to DG methods, see [24, 28, 22, 23]. The stabilization techniques mentioned so far implicitly introduce artificial dissipation. On the other hand, when spectral methods are applied to hyperbolic conservation laws we find directly introduced spectral viscosity terms. These diffusion-like terms depend on the spatial resolution and distinguish different frequencies of the numerical solution in order to stabilize the calculation without losing the high order approximation properties of the scheme. The concept of spectral vanishing viscosity for non-linear scalar conservation laws was introduced to spectral methods in [26]. Subsequently, the viscosity term has often been implemented as a spectral filter, see for example [5, 6], resulting in higher computational efficiency. Considering DG methods, there are other formulations that incorporate artificial viscosity terms as well, such as [14, 21, 8]. As the usual second order artificial viscosity is employed there – combined with a discontinuity indicator for an adaptive damping in [21, 8] – the high order approximation might again be lost in regions close to discontinuities. In addition, the regular diffusion term has to be incorporated in the space discretization, e.g. by adding stabilizing terms as in [14, 8] that take account for potentially discontinuous solution derivatives at interelement boundaries. The aim of spectral filtering is to stabilize the numerical scheme. Nevertheless, there will still be spurious Gibbs oscillations close to a discontinuity of the exact entropy solution. Reprojection techniques such as the Gegenbauer reconstruction method [11] provide sophisticated means to reconstruct a high order oscillation-free approximation even close to discontinuities but are difficult to apply in higher dimensions. First, an edge detection procedure is required to determine the regions of smoothness. In each of these regions a truncated Gegenbauer series is reconstructed using the oscillatory polynomial. To this end, two parameters have to be specified that greatly influence the quality of the approximation and for which an optimal choice is not known. As an alternative, in [25], Sarra studies the application of the digital total variation (DTV) filter developed by Chan, Osher and Shen [2] as a possible “black box” postprocessing tool for spectral methods. In the present paper, we derive a relation between modal filtering for discontinuous Galerkin methods on unstructured triangular grids and the concept of introducing spectral viscosity to the scheme. As a new development, the corresponding high order viscosity term is based on the Sturm-Liouville operator associated with the Dubiner polynomials and implemented as a spectral filter modifying the coefficients of the Dubiner expansion of the numerical solution. This formulation has the advantage of a reduced computational cost compared to limiters, (H)WENO reconstruction or the introduction of regular diffusion and may nevertheless serve as a stabilizing mechanism for solutions having strong shocks. The connection between spectral viscosity and spectral filtering, which is undertaken for the first time in the context of DG methods on unstructured grids, furthermore allows to specify conditions on the filter strength with respect to time step choice and mesh refinement. In the case of hyperbolic problems, we obtain invariance of the filter under mesh refinement for the usual choice of a constant ratio of time step and mesh width. In order to remove the remaining oscillations we apply the image processing technique of digital total variation filtering introduced by Chan, Osher and Shen in the new context of DG solutions. Numerical experiments for a two-dimensional linear advection equation confirm the designed order of 2
accuracy of the scheme for polynomial degrees up to 8 while an investigation of the influence of global and adaptive spectral filtering clearly shows the need of an adaptive procedure. The spectral and DTV filtering techniques are tested for a nonlinear conservation law with discontinuous solution. This paper is organized as follows. In section 2 we describe the discontinuous Galerkin space discretization used as a basic scheme while high order filters as well as the spectral viscosity method as stabilizing techniques are introduced in section 3. Afterwards, we transfer the spectral viscosity modification to the DG method and present the resulting weak filter. Section 4 deals with the digital total variation filter for which we prove conservativity in the limit. Subsequently, numerical experiments are presented concerning the influence of the weak filter on the order of convergence as well as the application of spectral and DTV filtering to a non-linear equation with discontinuous solution. The last section contains a conclusion as well as an outlook on future work.
2
The Discontinuous Galerkin Space Discretization
We consider two-dimensional scalar hyperbolic conservation laws of the form ∂ u(x, t) + ∇x · f (u(x, t)) = 0, ∂t
(x, t) ∈ Ω × R+ ,
(1)
where Ω ⊂ R2 is an open polygonal domain in which initial conditions u(x, 0) = u0 (x) are given. We further assume appropriately posed boundary conditions which respect the characteristic directions. The discontinuous Galerkin discretization in space is now obtained as follows. Let T h be a conforming triangulation of the closure Ω of the computational domain and let V h be the piecewise polynomial space defined by V h = {vh ∈ L∞ (Ω) | vh |τi ∈ P N (τi ) ∀ τi ∈ T h }, where P N (τi ) denotes the space of all polynomials on τi of degree ≤ N . In general, N may take different values on different elements. In this paper, focusing on the development of high order damping techniques, we simplify to a fixed polynomial degree on the computational domain Ω. Multiplying equation (1) by test functions in V h , integrating over Ω and using the divergence theorem leads to the semidiscrete equation Z Z X Z d u vh dx + f (u) · n vh dσ − f (u) · ∇x vh dx = 0, ∀vh ∈ V h . (2) dt Ω ∂τ τ i i h τi ∈T
We then strive for an approximation uh : Ω × R+ → R of u, that satisfies equation (2) as well as uh (·, t) ∈ V h for any t ∈ R0+ . As (2) is linear in vh , for each triangular subset τi ∈ T h it is sufficient to consider only those test functions vh vanishing outside τi and we obtain the local discretisation Z Z Z d uh Φ dx + f (uh ) · n Φ dσ − f (uh ) · ∇x Φ dx = 0, (3) dt τi ∂τi τi 3
valid for any triangular subset τi ∈ T h and any polynomial Φ ∈ P N (τi ). Since the approximation uh (·, t) is polynomial on τi , we may consider (3) as a system of ordinary differential equations for the time-dependent coefficients of uh corresponding to a suitable basis of P N (τi ). For spectral methods on triangular grids, in [7], M. Dubiner described a basis of orthogonal polynomials on a reference triangle T using the one-dimensional Jacobi polynomials Pnα,β . These two-dimensional polynomials as well as their properties and use for spectral methods are also described in detail in [16]. On the reference element T = {(r, s) ∈ R2 | − 1 ≤ r, s; r + s ≤ 0} the Dubiner polynomials are given by 1 − s l 1+r 2l+1,0 −1 Pm (s), l, m ∈ N0 , (4) Φlm (r, s) = Pl0,0 2 1−s 2 and the set {Φlm | 0 ≤ l + m ≤ N } represents a basis of P N (T ). Denoting by Λi : τi → T, x 7→ Ai x + bi , where Ai ∈ R2×2 and bi ∈ R2 , an orientationpreserving affine transformation which maps the specific triangle τi to the reference element T , we obtain a basis of P N (τi ) consisting of the polynomials Φlm ◦ Λi , 0 ≤ l + m ≤ N . Thus, uh |τi can be expanded as X uh (Λ−1 u ˆilm (t)Φlm (r, s) i (r, s), t) = l+m≤N
and exploiting orthogonality yields T uh (Λ−1 ˆilm (t), where i (r, s), t) Φlm (r, s) drds = γlm u γlm = kΦlm kL2 . Therefore, equation (3) written in the Dubiner coefficients gives Z Z 2 d i T f (uh ) · n (Φlm ◦ Λi ) dσ + f (uh ◦ Λ−1 (5) u ˆ =− i ) · Ai ∇r,s Φlm drds, dt lm γlm |τi | ∂τi T R
for 0 ≤ l, m ≤ N , where we employed the transformation of spatial derivatives to the reference element ∇x = ATi ∇r,s as well as the fact that the determinant of the Jacobian A−1 of the map Λ−1 is equal to |τi |/2, denoting by |τi | the area of the triangle τi . i i We now approximate the integrals in (5) by quadrature formulae which shall be exact for polynomials of degree 2N in the reference element and for polynomials of degree 2N + 1 on each part of the cell boundaries ∂τi respectively, in order to obtain a truncation error of order N + 1 in (5), see [3]. To this end, we employ Gauss quadrature at the interelement and domain boundaries while a high order quadrature rule for the elements is constructed similar to [16] by using a singular transformation of the reference triangle to the square [−1, 1]2 and one-dimensional Gauss quadrature rules. The resulting quadrature points for the volume integral are completely located whithin the interior of T . As the functions uh ∈ V h may be discontinuous at the cell interfaces we have to give a precise meaning to the expression f (uh ) · n in equation (5) above. To this end, at the Gauss points on edges, we employ a numerical flux function H which depends on the physical quantities uL and uR as well as the normal vector n and is assumed to fulfill the following properties. • H is defined for each argument (uL , uR , n) ∈ R × R × B1 , where B1 = {n | knk2 = 1}, and is Lipschitz continuous with respect to uL and uR . • H is consistent with the flux vector f , i.e. H(u, u, n) = f (u) · n, 4
u ∈ R, n ∈ B1 .
• H is conservative: H(uL , uR , n) = −H(uR , uL , −n),
uL , uR ∈ R, n ∈ B1 .
• H is non-decreasing in the first argument and non-increasing in the second one. In our computations for scalar conservation laws we use the Godunov flux given by min f (u) · n if uL ≤ uR , uL ≤u≤uR H(uL , uR , n) = max f (u) · n otherwise. uR ≤u≤uL
In order to include the quadrature rules and the numerical flux function into the semidiscrete system, we need to introduce the following notation. Let ξν ∈ [−1, 1], ν = 1, . . . , nedge denote the Gaussian integration points with corresponding weights ων and let (rµ , sµ ) ∈ T, µ = 1, . . . , ninner be the integration points on T with associated weights ω ˜ µ . Furthermore, let ∂τi = ∪3j=1 Γij be the decomposition of ∂τi into straight edges Γij with length |Γij | and normal vector nij and let xij : [−1, 1] → Γij be the affine transformation mapping the Gaussian points from [−1, 1] to Γij . If there is an element τk ∈ T h with Γij = ∂τi ∩ ∂τk we denote the index k by n(i, j). For the boundary treatment we decompose ∂Ω into an inflow and an outflow boundary part as ∂Ω = Γin ∪ Γout . The semidiscrete system for the coefficients u ˆilm is then given by nedge 3 X d i 1 X |Γij | ων H uih (xij (ξν ), t), uij (x (ξ ), t), n u ˆlm = − ij ν ij Φlm (Λi (xij (ξν )) h dt γlm |τi | j=1
+
nX inner
ν=1
T ω ˜ µ f (uh ◦ Λ−1 i (rµ , sµ )) · Ai ∇r,s Φlm (rµ , sµ ),
(6)
µ=1
where we employ the short notation uih into uij h by setting n(i,j) uh ij uh = u|Γij uih
= uh |τi and incorporate the boundary conditions if Γij = ∂τi ∩ ∂τn(i,j) , if Γij ⊂ Γin , if Γij ⊂ Γout .
We thus obtain a system of ordinary differential equations for the coefficients u ˆilm , which may be solved by appropriate time integration schemes depending on the specific application under consideration.
3 3.1
The Filtering Procedure High Order Filters
The most efficient way of introducing artificial viscosity to spatially high order methods while maintaining the approximation order is modifying the expansion coefficients via a high 5
order filter. While for spectral element methods demanding a continuous approximation throughout the computational domain the spectral filters must be modified in order to respect interelement continuity, this is not necessary in the case of DG methods leading to a very simple scheme. Definition 1 For an integer p ≥ 1 we define a filter of order p as an even function σ ∈ C p−1 (R) with the properties σ(0) = 1, (l)
σ (0) = 0,
(7) l = 1, 2, . . . , p − 1.
(8)
If a filter of order p satisfies the additional condition σ(η) = 0,
|η| ≥ 1,
(9)
the regularity condition σ ∈ C p−1 (R) yields the property σ (l) (1) = 0
for 0 ≤ l ≤ p − 1 .
(10)
The asymtotic analysis for filtered Fourier expansions has been rigorously carried out by Vandeven in [27]. Given a periodic, piecewise smooth function u : R → R and a filter σ of order p satisfying the additional condition (9), he proved the following error bounds for the filtered partial sum X k σ u ˆk eikx , uN (x) = σ N |k|≤N
where uN (x) =
P
|k|≤N
u ˆk eikx is the truncated Fourier expansion of u(x).
Theorem 1 1) Let u ∈ C p (R). For x ∈ R the error |u(x) − uσN (x)| is bounded by |u(x) − uσN (x)| ≤ C1 ·
1 N p−1/2
,
where C1 is a constant independent of x and N . 2) If u possesses one or more jump discontinuities and is continuous at x ∈ R, we have |u(x) − uσN (x)| ≤ C2 ·
1 , d(x)p−1 N p−1
where d(x) denotes the distance from x to the nearest discontinuity and C2 is a constant independent of x and N . A very popular choice, though not possessing property (10), is the exponential filter σ(η) = exp(−αη p ), with α = − ln(eps), where eps is the machine accuracy. 6
(11)
As already remarked by Vandeven, the approximation results in Theorem 1 also apply to Chebyshev series since the Chebychev coefficents of a function v : [−1, 1] → R are proportional to the Fourier coefficients of u(θ) = v(cos θ). However, there are only partial results for Legendre series, see [13], and to the authors’ knowledge there are no results concerning the approximation properties of filtered Dubiner expansions. Although the analytic investigation of modal filters applied to Dubiner polynomials is beyond the scope of this article, we will show by numerical results, that these filters may be used to stablize nonlinear conservation laws with discontinuous solutions.
3.2
The Spectral Viscosity (SV) Method
In [26], Tadmor showed that spectral methods applied to conservation laws may fail to converge to the exact entropy solution as they lack sufficient entropy dissipation. He therefore introduced the spectral vanishing viscosity and super viscosity methods, adding to the spectral method a diffusion term of increasing strength with growing wave numbers. In this subsection, we describe the SV method for periodic scalar equations ∂ ∂ u+ f (u) = 0 on [−π, π] × [0, T ], (12) ∂t ∂x as well as its relation to high order filters. The original SV version was developed for the Fourier method, where the Fourier projection, Z π X 1 u ˆ(k, t)eikx , u ˆ(k, t) = u(x, t)e−ikx dx, PN u(x, t) = 2π −π |k|≤N
of the exact entropy solution u(x, t)P of equation (12) is approximated by an N -trigonometric polynomial of the form uN (x, t) = |k|≤N u ˆk (t)eikx . Starting with the projection of the given initial conditions, i.e. uN (x, 0) = PN u(x, 0), the approximation is evolved according to the semidiscrete equation ∂ ∂ uN + PN f (uN ) = 0. ∂t ∂x
(13)
The derivation of the Dubiner filter is related to the spectral viscosity modification of (13) in the so-called super viscosity form ∂ ∂ 2p ∂ uN + PN f (uN ) = N (−1)p+1 2p uN , ∂t ∂x ∂x
(14)
which is also advocated in [10]. The parameter N representing the viscosity strength hereby depends on the approximation order as N ∼
Cp , N 2p−1
(15)
where the constant Cp may be chosen by Cp ∼
p X
|f |C k kuN kk−1 L∞ ,
k=1
7
(16)
with |f |C k = k∂uk f (u)kL∞ . As shown in [10], this method can be implemented as a high order filter by the subsequent considerations. Solving (14) by a splitting method leads to the following scheme on the time interval t ∈ [tn−1 , tn ]: ∂ 2p ∂ wN = N (−1)p+1 2p wN , ∂t ∂x ∂ ∂ uN + PN f (uN ) = 0, ∂t ∂x
wN (x, tn−1 ) = uN (x, tn−1 ),
(17)
uN (x, tn−1 ) = wN (x, tn ).
(18)
Hence, before employing the basic spectral scheme (13), the damping step (17) is carried out. This first step is equivalent to filtering in Fourier space since for the Fourier coefficients it reads d w ˆk = −N k 2p w ˆk , 0 ≤ |k| ≤ N. dt Recalling N = Cp /N 2p−1 , the filtered coefficients are then given by 2p ! k σ w ˆk = exp −Cp N ∆t w ˆk , N P k σ = ˆk eikx , with σ(η) = exp −Cp N ∆tη 2p . Hence, after each time i.e. wN |k|≤N σ N w step associated with the spectral scheme, an exponential filter (11) of order 2p including a filter strength α = Cp N ∆t is applied to uN (x, tn ) = wN (x, tn ). For global Fourier spectral methods, the CFL condition for ∆t leads to a constant value of N ∆t, see [10]. Therefore, in this case it is possible to choose a constant α for growing values of N . Subsequently, the SV method has been extended to Legendre [19] and ChebyshevLegendre [20] methods, where the time step has to scale like ∆t = O(N −2 ), as shown in [12]. Thus, for these expansions the choice of α must depend on N as αN = O(1/N ).
3.3
Filtering the Dubiner Expansion
For the two-dimensional equation (1), we use the basic DG scheme described in section 2 whereby a high order filter is applied to the Dubiner coefficients after each time step. This filter is obtained by an extension of the SV method to multidomain Dubiner expansions. Thus, the implemented damping technique can be understood as resulting from an elementwise spectral viscosity formulation even thought directly applied to the Dubiner coefficients. Considering the Sturm-Liouville problem associated with the Dubiner basis and transferring the conditions on the viscosity strength N in (15) to the elementwise discretization, the spectral viscosity modification of (1) is obtained as follows. The Dubiner polynomials satisfy the singular Sturm-Liouville equation Lr,s Φ(r, s) + λlm Φ(r, s) = 0 ,
(r, s) ∈ T,
(19)
with the differential operator ∂ ∂ ∂ ∂ ∂ ∂ Lr,s = (1 + r) (1 − r) − (1 + s) + (1 + s) (1 − s) − (1 + r) ∂r ∂r ∂s ∂s ∂s ∂r 8
and λlm = (l + m)(l + m + 2), see [16]. As already considered in [10] for Chebyshev- and Legendre expansions, the operator ∂x2p employed in (14) is not the only one qualifying for the construction of viscosity terms. On the reference element T , we may as well consider high order operators of the form (Lr,s )p . Aiming at a two-dimensional analogon of equation (17) on T , for each triangular subset τi ∈ T h we consider a function wh,i : T × [tn−1 , tn ] → R, satisfying wh,i (·, t) ∈ PN (T ) for any time t ∈ [tn−1 , tn ], and define the initial data at the beginning of the time intervall p by wh,i (r, s, tn−1 ) = uh Λ−1 i (r, s), tn−1 . Applying the operator (Lr,s ) to the Dubiner expansion of wh,i and using (19) we obtain X p i ˆlm (Lr,s )p wh,i = (−1)p λlm w Φlm , l+m≤N
which can be implemented at low computational cost. Recalling the definition of the constant Cp in (16), we find that the viscosity strength N depends on the characteristic speed associated with the flux vector f . To understand the influence of this aspect in our context we transform the equation ∂ uh + ∇x · PN f (uh ) = 0, ∂t
x ∈ τi , t ∈ [tn−1 , tn ],
where PN now denotes the componentwise Dubiner projection, to the reference element coordinates via the map Λi (x) = Ai x + bi . Using ∇x = ATi ∇r,s , we obtain ∂ −1 uh (Λ−1 i (r, s), t) + ∇r,s · PN (Ai f (uh (Λi (r, s), t))) = 0, ∂t
(20)
which is solved by the basic DG scheme described in section 2. The corresponding filter step is given by ∂ wh,i = N (−1)p+1 (Lr,s )p wh,i , ∂t
(21)
with viscosity strength N =
C˜p , hi (N + 1)2p−1
(22)
as in (15), except for the newly introduced length measure hi taken as the shortest distance of the barycenter of τi to the element boundary ∂τi . The above definition of N is closely related to the choice of Cp in the global SV method since for d ∈ R+ the definition of Cp in (16) yields Cp (d · f ) = d · Cp (f ). In order to obtain an elementwise scheme which is consistent with the global method, we derive from (20) that N should scale linearly with det Ai . However, as det Ai is proportional to 1/hi , this factor has to be introduced in the definition of N . Expanding wh,i in Dubiner polynomials, equations (21) and (22) yield C˜p C˜p (N + 1) l + m 2p i d i p p i w ˆ =− (l + m) (l + m + 2) w ˆlm ≈ − w ˆlm dt lm h(N + 1)2p−1 h N +1 i , 0 ≤ l + m ≤ N . Therefore, an exponential filter for the Dubiner coefficients w ˆlm
σ(η) = exp(−αη 2p ),
η=
l+m , N +1 9
α=
C˜p (N + 1)∆t h
(23)
of order 2p is applied to the Dubiner expansion. Remark: In the case of explicit time integration schemes solving the semidiscrete equation (20), the usual time step restriction for (20) requires ∆t = O(hi /N 2 ), see [16]. An accordingly chosen time step hence causes the filter σ(η) in (23) to be invariant under mesh refinement and α to scale with N according to α = O(1/N ).
4
The Digital Total Variation Filter
DTV filtering applies to general graphs [V, E] with a finite set of nodes V and edges E. If two nodes α, β ∈ V are linked by an edge, we write α ∼ β. For a given node α ∈ V we further denote by Nα = {β ∈ V | α ∼ β} the set of nodes linked to α. Let u0α , α ∈ V, be the oscillatory nodal values of uh (·, tout ) at time tout where a truthful pointwise solution is desired and denote by u0 the vector with components u0α . Following [2] the DTV filter is implemented as an iterative procedure X hαβ (un )unβ + hαα (un )u0α , n = 0, 1, . . . , (24) un+1 = α β∈Nα
where the filter coefficients are given by hαβ (u) =
ωαβ (u) X , λ+ ωαγ (u)
hαα (u) =
λ X
λ+
γ∈Nα
,
(25)
ωαγ (u)
γ∈Nα
for appropriate non-negative weights ωαβ measuring the local variation of the given data and a non-negative, user-dependent parameter λ that balances the competing tasks of removing spurious oscillations and retaining relevant information of the noisy initial data. The weights in (25) are chosen by ωαβ (u) = |∇α1u|a + |∇β1u|a , where 1/2
|∇α u|a =
X
(uβ − uα )2 + a2
β∈Nα
is the regularized local variation at node α equipped with a small regularization parameter a > 0 to avoid a zero denominator. The DTV filter has been constructed with a discrete variational problem in mind. It was shown in [2] that the above choice of weights results from the variational problem of minimizing the penalized total variation energy X
|∇α u|a +
α∈V
λ ku − u0 k22 2
depending on the given parameters a, λ and the initial noisy data u0 . A postprocessing technique for approximate solutions to conservation laws should not violate the conservative properties of the numerical scheme. Hence, we show that if the DTV filtering procedure converges it is conservative in the limit and may thus safely be used. 10
Lemma 2 If the iterative method (24) withX filter coefficients determined by (25) converges, X n 0 the limit u = lim u satisfies the equation uα = uα , i.e. in the limit the DTV filter n→∞
α∈V
α∈V
is conservative. In addition, if λ = 0, each filtering step is conservative. Proof: X We first consider the case λ 6= 0. Since u is a fix point of (24), multiplying by λ+ ωαγ (u) γ∈Nα
yields
λ +
X
ωαγ (u) uα =
γ∈Nα
X
ωαβ (u)uβ + λu0α ,
∀α ∈ V,
β∈Nα
which can be rearranged to X ωαβ (u)(uβ − uα ) + λ(u0α − uα ) = 0,
∀α ∈ V.
β∈Nα
Taking the sum over the nodes in V and exploiting symmetry of the weights with respect to the node indices, i.e. ωαβ (u) = ωβα (u), the first sum in the above equation vanishes and we obtain X λ· (u0α − uα ) = 0. α∈V
which gives the desired result as λ 6= 0. For λ = 0, using
X
hβα (un ) = 1 and hαβ (un ) = hβα (un ), we obtain
α∈Nβ
X α∈V
un+1 = α
X X
hαβ (un )unβ =
α∈V β∈Nα
X β∈V
unβ
X
hαβ (un ) =
α∈Nβ
X
unβ ,
β∈V
i.e. discrete conservation in each iteration step.
5 5.1
Numerical Experiments Experimental Order of Convergence for Smooth Solutions
As first test case for a study on the numerical order of convergence, we consider the linear transport ∂ u(x, t) + ∇x · u(x, t) = 0, ∂t
(x, t) ∈ (0, 1)2 × (0, T ]
of the scaled Gaussian kernel u0 (x) = u(x, 0) = 0.2e−500((x1 −0.2)
11
2 +(x
2 −0.3)
2)
(26)
across an unstructured triangular grid. The boundary conditions are appropriately defined with respect to the exact solution u(x1 , x2 , t) = u0 (x1 − t, x2 − t). Table 3 and Figure 4 represent a validation of the DG method for (26) at a final time of tout = 0.5. The solutions have been computed on a hierarchy of green-refined unstructured triangular grids. The smallest grid is shown in Figure 1 and consists of K = 296 triangles while the subsequent grids have K = 1184, K = 4736 and K = 18944 elements, respectively. For the time integration we used a 4th order low storage Runge-Kutta method with a time step of ∆t = 10−5 which was chosen small enough to investigate the spatial high order of the scheme. Table 3 lists the error measured in the global L2 norm and the experimental order of convergence for polynomial degrees of 3 ≤ N ≤ 8 while in Figure 4 the L2 error is plotted against the number of triangles of the respective grid to depict the convergence rate. In all cases the influence of the time integration can be neglected and the spatial order of the scheme is preserved - up to the finite precision destroying the convergence order for K = 18944 and N = 8. Figure 2 depicts the initial data as well as the computed solution at tout = 0.5 for K = 296 and N = 5.
Figure 1: Smallest grid consisting of 296 triangles.
Figure 2: Initial data and computed solution at tout = 0.5 for N = 5.
12
Figure 3: Experimental order of convergence for the linear transport equation. Global L2 -error EOF N Global L2 -error EOF N 1.171084e-03 3.521460e-04 6.818670e-05 4.102210e+00 7.670696e-06 5.520672e+00 3 4 2.774005e-06 4.619448e+00 2.138201e-07 5.164888e+00 1.681435e-07 4.044205e+00 6.559287e-09 5.026715e+00 7.996430e-05 2.044012e-05 1.059437e-06 6.237986e+00 1.311833e-07 7.283676e+00 5 6 1.540299e-08 6.103944e+00 1.041515e-09 6.976757e+00 2.389907e-10 6.010112e+00 8.228855e-12 6.983776e+00 4.142200e-06 1.007003e-06 1.855943e-08 7.802101e+00 2.157085e-09 8.866769e+00 7 8 7.270799e-11 7.995823e+00 4.558819e-12 8.886207e+00 3.075400e-13 7.885196e+00 1.279585e-13 5.154912e+00
Figure 4: L2 error in mesh refinement. K is the number of triangles. Applying the high order filter (23) given in section 3 to this problem severely degrades the spatial accuracy which can be seen in Table 1 listing the global L2 error for 3 ≤ N ≤ 6. The experimental order of convergence between 1.2 and 2 obviously does not correspond to the designed order of the scheme. Therefore, an ad hoc indicator is implemented to decide on which elements filtering has to be applied. This indicator is inspired by the sub-cell resolution strategy in [21], which has not been applied to spectral filtering before. In our computations we employ a fourth order filter and choose the filter strength αN,i on τi as 0 if ωi < (N + 1)−4 , αN,i = 0.5 · N −1 else, !−1 X X i i where the indicator ωi is given by ωi = u ˆlm · u ˆlm and thus based on the l+m=N
decay rate of the calculated Dubiner coefficients. 13
l+m
