Journal of Computational Physics 349 (2017) 97–121
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Journal of Computational Physics www.elsevier.com/locate/jcp
Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws Lin Fu, Xiangyu Y. Hu ∗ , Nikolaus A. Adams Institute of Aerodynamics and Fluid Mechanics, Technische Universität München, 85748 Garching, Germany
a r t i c l e
i n f o
Article history: Received 18 October 2016 Received in revised form 1 March 2017 Accepted 30 July 2017 Available online 3 August 2017 Keywords: TENO WENO LES DNS Spectral property High-order scheme
a b s t r a c t In this paper, we extend the range of targeted ENO (TENO) schemes (Fu et al. (2016) [18]) by proposing an eighth-order TENO8 scheme. A general formulation to construct the high-order undivided difference τ K within the weighting strategy is proposed. With the underlying scale-separation strategy, sixth-order accuracy for τ K in the smooth solution regions is designed for good performance and robustness. Furthermore, a unified framework to optimize independently the dispersion and dissipation properties of highorder finite-difference schemes is proposed. The new framework enables tailoring of dispersion and dissipation as function of wavenumber. The optimal linear scheme has minimum dispersion error and a dissipation error that satisfies a dispersion-dissipation relation. Employing the optimal linear scheme, a sixth-order TENO8-opt scheme is constructed. A set of benchmark cases involving strong discontinuities and broadband fluctuations is computed to demonstrate the high-resolution properties of the new schemes. © 2017 Elsevier Inc. All rights reserved.
1. Introduction The weighted essentially non-oscillatory (WENO) scheme first proposed by Liu et al. [1], and improved by Jiang and Shu [2], is a reliable numerical approach for solving linear and nonlinear Partial Differential Equations (PDE). Nevertheless, there are several issues that require consideration, e.g. the accuracy order may degenerate near critical points [3], WENO schemes typically are unnecessarily dissipative at small scales [4], and stability problems may occur for very-high-order variants [5]. Henrick et al. [3] first recognized the order degeneration of WENO-JS [2] near critical points and proposed WENO-M to fix this problem. Later, Borges et al. [6] proposed a new weighting strategy to improve the order degeneration problem by introducing a high-order global smoothness indicator. Other proposals were to adjust the sensitivity parameter as a function of x [7]. More recently, Acker et al. [8] state that increasing the contribution of the candidate stencils which are less smooth is more important than improving the accuracy at critical points [6]. Based on the observation that the effective numerical dissipation is directly relevant to the nonlinear adaptation of WENO schemes, Hill and Pullin [4] proposed to freeze the adaptation when the ratio between the largest and the smallest calculated smoothness indicator is less than a problem dependent threshold [9]. By increasing the constant parameter C in the weighting strategy to reduce the sensitivity of the nonlinear adaptation, Hu and Adams [10] [11] derived the sixth-order adaptive central-upwind WENO-CU6 scheme and the scale-separation WENO-CU6-M scheme. In [8], a new WENOZ+ scheme
*
Corresponding author. E-mail addresses:
[email protected] (L. Fu),
[email protected] (X.Y. Hu),
[email protected] (N.A. Adams).
http://dx.doi.org/10.1016/j.jcp.2017.07.054 0021-9991/© 2017 Elsevier Inc. All rights reserved.
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L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
which incorporates an extra term in the weighting formula to correct the under-weighting of less-smooth candidate stencils has been proposed. The WENOZ+ scheme generally shows similar performance as WENO-Z for shock-dominated problems. Another approach to dissipation reduction is to optimize the spectral resolution properties of the corresponding linear scheme. By introducing a downwind stencil, optimized third- and fourth-order WENO schemes with low dissipation have been constructed, based on the six- and eight-point full stencil [12] [13]. A drawback is that anti-dissipation may be inherently built in for a certain wavenumber range and that the accuracy order cannot be adjusted flexibly [9]. For linear finite-difference schemes with 2r coefficients, the dispersion and dissipation properties can be determined independently by two individual parameters while maintaining (2r − 2)th-order of accuracy, leading to fourth- and sixth-order hybrid MDCD schemes developed by Ren et al. [14] [15]. However, numerical experiments reveal that the sixth-order MDCD6-HY shows no significant improvement over WENO-CU6 except for higher computational efficiency due to the hybridization [15]. For very-high-order WENO schemes, achieving numerical stability becomes difficult due to the large candidate-stencil width. Specific methods, such as the monotonicity-preserving method [16] and the order-reduction method [5], can enhance robustness. For very-high-order WENO-Z schemes, the problem can be remedied by adjusting the power parameter in the weighting strategy to r − 1 for (2r − 1)th-order schemes [17]. However, the induced nonlinear numerical dissipation at intermediate- and high-wavenumbers significantly degenerates the performance. Recently, a family of high-order targeted ENO (TENO) schemes, which feature properties that address the main drawbacks of WENO schemes, has been proposed by Fu et al. [18]. Main properties of TENO schemes are: (i) Arbitrarily high-order odd (upwind) or even (central) schemes can be constructed within a unified framework. The scheme can gradually reduce to third-order reconstruction when there are multiple discontinuities close to each other. Numerical robustness is improved even for high-order versions. (ii) The order-degeneration problem is avoided, and spectral resolution properties of a nonlinear TENO scheme recover to that of its corresponding linear scheme for low to intermediate wavenumbers. Moreover, the nonlinear dissipation is controllable by an ENO-like stencil selection procedure. (iii) Spectral resolution properties can be optimized to satisfy a dissipation-dispersion relation with accuracy order reduction by one. However, three important issues have not been addressed in [18]: (i) a general expression to construct τ K in the weighting strategy for arbitrarily high-order TENO schemes is missing. (ii) The dispersion error cannot be optimized and may even increase for spectrally optimized odd-order schemes. (iii) The performance of very-high-order TENO schemes in terms of accuracy and robustness currently is unexplored. In this paper, we propose a procedure to formulate very-high-order TENO schemes and assess the performance of two specific examples. A general formulation to construct the high-order undivided difference in the weighting strategy is proposed. A unified framework to optimize dispersion and dissipation properties independently is developed. Within this framework, a family of finite-difference schemes, for which the dispersion and dissipation properties are tailored to a specific relation, can be constructed. The eight-point TENO8 scheme with optimum accuracy and the TENO8-opt scheme with optimum spectral properties are formulated and analyzed. The remainder of this paper is organized as follows. In section 2, the concept and formulation of TENO schemes are reviewed, and the eighth-order TENO8 scheme is constructed. A general formulation for an undivided difference with sixthorder accuracy in the weighting strategy is proposed. In section 3, a unified framework to optimize the dispersion and dissipation property of finite-difference schemes separately is proposed and discussed in detail. The spectral properties of the eighth-order TENO8 scheme and the optimized sixth-order TENO8-opt scheme are analyzed by the ADR analysis. In section 4, a set of benchmark cases involving strong discontinuities and rich scales are carried out. Concluding remarks are summarized in the last section. 2. Very-high-order targeted ENO schemes In this section, we first review the concept of targeted ENO schemes. Then, the eighth-order TENO8 scheme is constructed and a general formulation to construct the sixth-order undivided difference τ K is proposed. 2.1. Concepts of TENO schemes We first consider the one-dimensional hyperbolic conservation law
∂ ∂u + f ( u ) = 0, ∂t ∂x
(1)
where u denotes the conservative variable, f denotes the flux function and the characteristic velocity is assumed to be ∂ f (u ) positive ∂ u > 0. The discretization on a uniform mesh, e.g. xi = i x where x denotes the grid spacing, results in a system of ordinary differential equations
du i dt
=−
∂ f x=xi , i = 0, · · · , n. ∂x
(2)
In a general semi-discrete form, Eq. (2) can be approximated by a conservative finite-difference scheme as
du i dt
=−
1
x
(hi +1/2 − hi −1/2 ),
(3)
L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
99
Fig. 1. Candidate stencils with incremental width for high-order reconstruction.
where the primitive function h(x) is implicitly defined by
f (x) =
x+ x/2
1
x
h(ξ )dξ .
(4)
x−x/2
Eq. (3) can be approximated as
du i dt
≈−
1
x
( f i +1/2 − f i −1/2 ),
(5)
where the numerical fluxes ˆf i ±1/2 are computed from a convex combination of K − 2 candidate-stencil fluxes
f i +1/2 =
K −3
w k f k,i +1/2 ,
(6)
k =0
for K -point reconstruction. To obtain a K th-order approximation for f i +1/2 , a (rk − 1)-degree interpolation on each candidate stencil leads to
h(x) ≈ ˆf k (x) =
r k −1
al,k xl ,
(7)
l =0
where rk denotes the stencil width of candidate flux k. After substituting Eq. (7) into Eq. (4) and evaluating the integral functions at the stencil nodes, the coefficients al are determined by solving the resulting system of linear algebraic equations. When using the monomial basis Eq. (7), the resulting linear system for determining the interpolation coefficients can be ill-conditioned for very high-order polynomials. In this case, Newton interpolation based on divided-difference tables should be used instead [19]. For the schemes developed in this paper, the latter is not necessary. The direct introduction of linear weights in Eq. (6) typically leads to Gibbs oscillations [20] near discontinuities. The essential difference of the nonlinear-adaptation strategy in a nonlinear scheme distinguishes ENO [21], WENO [2] and TENO [18]. 2.1.1. Candidate stencils with incremental width As shown in Fig. 1, unlike the classical WENO scheme which assembles candidate stencils of same width, high-order reconstruction is achieved by combining low-order stencils of width. K th-order schemes, odd (upwind) or Kincremental −3 even (central), can be constructed by the stencil combination k=0 S k . The sequence of stencil widths r for the low-order candidate stencil k is given as
{rk } =
⎧ K +2 ⎪ ⎪ }, if mod ( K , 2) = 0, { 3, 3, 3, 4, · · · , ⎪ ⎪ 2 ⎪ ⎪ ⎨ k=0,··· , K −3
K +1 ⎪ ⎪ }, if mod ( K , 2) = 1. { 3, 3, 3, 4, · · · , ⎪ ⎪ 2 ⎪ ⎪ ⎩
(8)
k=0,··· , K −3
According to the smoothness measurement of local flow features, the TENO reconstruction degenerates gradually to 3rd-order, and is able to recover the robustness of the classical fifth-order WENO-JS scheme [18].
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2.1.2. Scale separation The scale-separation formulation is designed to isolate discontinuities from smooth regions by
γk = C +
q
τK
, k = 0, · · · , K − 3 ,
βk,r + ε
(9)
where ε = 10−40 is introduced to avoid division by zero, and τ K denotes the high-order undivided difference on the full stencil. For WENO schemes, a large value of C improves the performance in smooth regions but degrades the shockcapturing capability [11] while the tendency is reversed in terms of the integer q [6]. Therefore, it is difficult to achieve sufficient scale separation. With TENO schemes, this is accomplished by increasing q up to 6 and decreasing C to 1. In this way, the discontinuity-detection capability is enhanced and simultaneously low dissipation at low and intermediate wave-numbers is achieved by a tailored ENO-like stencil selection procedure. Following Jiang and Shu [2], βk,r can be evaluated as
βk,r =
r −1
xi +1/2 2 j −1
x
j =1
xi −1/2
dj dx j
2 ˆf k (x)
dx.
(10)
2.1.3. ENO-like stencil selection In order to restore the ENO-property near discontinuities, the smoothness indicator is first normalized as
γ χk = K −k3 k =0
γk
(11)
,
and then filtered by a sharp cutoff function
δk =
if χk < C T , otherwise.
0, 1,
(12)
The basic idea is that one candidate stencil is completely removed from the reconstruction when a discontinuity is detected on this stencil, otherwise it is included with the optimal weight. The choice of C T depends on a compromise between good spectral properties and numerical robustness for discontinuity capturing. 2.1.4. High-order reconstruction scheme In order to remove the contributions from candidate stencils containing discontinuities, the optimal weights dk subjected to the cut-off δk are re-normalized as
dk δk w k = K −3 . k=0 dk δk Afterwards, the K th-order reconstructed numerical flux evaluated at cell face i +
ˆf K
i +1/2
=
K −3
(13) 1 2
is assembled as
w k ˆf k,i +1/2 .
(14)
k =0
2.2. Eighth-order TENO8 scheme For the eighth-order TENO8 scheme, the formulations of six candidate-stencil fluxes can be derived from Eq. (7) as
ˆf 0,i +1/2 = 1 (− f i −1 + 5 f i + 2 f i +1 ), 6
ˆf 1,i +1/2 = 1 (2 f i + 5 f i +1 − f i +2 ), 6 1
ˆf 2,i +1/2 = (2 f i −2 − 7 f i −1 + 11 f i ), ˆf 3,i +1/2 = ˆf 4,i +1/2 = ˆf 5,i +1/2 =
6 1
12 1 12 1 60
(3 f i + 13 f i +1 − 5 f i +2 + f i +3 ), (−3 f i −3 + 13 f i −2 − 23 f i −1 + 25 f i ), (12 f i + 77 f i +1 − 43 f i +2 + 17 f i +3 − 3 f i +4 ).
(15)
L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121 4 1 For maximum eighth-order accuracy, the optimal weights are d0 = 30 , d1 = 18 , d2 = 70 , d3 = 12 , d4 = 70 , d5 = 70 70 70 ble 2 in [18]. Explicit expressions in terms of cell-averaged quantities f i for the smoothness indicators are [7]
1
13
4 1
12
β0 = ( f i −1 − f i +1 )2 +
13
4 1
12 13
4 1
12
β2 = ( f i −2 − 4 f i −1 + 3 f i )2 + 36
see Ta-
( f i − 2 f i +1 + f i +2 )2 , ( f i −2 − 2 f i −1 + f i )2 ,
(−11 f i + 18 f i +1 − 9 f i +2 + 2 f i +3 )2 13
(2 f i − 5 f i +1 + 4 f i +2 − f i +3 )2 12 781 + (− f i + 3 f i +1 − 3 f i +2 + f i +3 )2 , 720 1 (−2 f i −3 + 9 f i −2 − 18 f i −1 + 11 f i )2 β4 = 36 13 + (− f i −3 + 4 f i −2 − 5 f i −1 + 2 f i )2 12 781 + (− f i −3 + 3 f i −2 − 3 f i −1 + f i )2 720 1 β5 = (−25 f i + 48 f i +1 − 36 f i +2 + 16 f i +3 − 3 f i +4 )2 144 13 + (35 f i − 104 f i +1 + 114 f i +2 − 56 f i +3 + 11 f i +4 )2 1728 781 + (−5 f i + 18 f i +1 − 24 f i +2 + 14 f i +3 − 3 f i +4 )2 2880 1 − (35 f i − 104 f i +1 + 114 f i +2 − 56 f i +3 + 11 f i +4 )( f i − 4 f i +1 + 6 f i +2 − 4 f i +3 + f i +4 ) 4320 32803 ( f i − 4 f i +1 + 6 f i +2 − 4 f i +3 + f i +4 )2 + 30240 +
5 , 70
( f i −1 − 2 f i + f i +1 )2 ,
β1 = (3 f i − 4 f i +1 + f i +2 )2 +
β3 =
101
(16)
2.3. High-order undivided difference τ K Following [3], Eq. (14) can be rearranged as
ˆf K
i +1/2 =
K −3
dk ˆf k,i +1/2 +
k =0
K −3
( w k − dk ) ˆf k,i +1/2 .
(17)
k =0
The first term on the right hand results in an order K scheme with standard weights. A sufficient condition for the overall scheme to restore order K is that the second term of Eq. (17) is order O (x K +1 ). Upon expanding this term, we obtain K −3
( w k − dk ) ˆf k,i +1/2 = hi +1/2
k =0
K −3
( w k − dk ) +
k =0
K −3
A k ( w k − dk )x3 +
k =0
K −3
( w k − dk )O(x4 ).
(18)
k =0
Due to renormalization Eq. (13), the first term on the right vanishes, so that the above condition becomes
( w k − dk ) = O(x K −2 ).
(19)
Without considering the effects of the sharp cutoff function, a more restrictive condition is
τK βk,r + ε
= O(x K −2 ).
As βk,r is order O (x2 ) in smooth regions,
(20)
τ K should satisfy the requirement of
τ K = O(x K ) in order to maintain the K th-order accuracy of the full-stencil scheme.
(21)
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Considering the ENO-like stencil selection, however, condition Eq. (20) reduces to a weaker criterion
τK βk,r + ε
= O(xs ),
s > 0.
(22)
Even at critical points, if the accuracy order of τ K is higher than that of βk,r , all γk converge to C q following Eq. (9) when x approaches zero; otherwise with a proper choice of C T , χk ≥ C T is also valid, so that the formal high-order accuracy can be recovered without degeneration. Although this less restrictive condition allows flexibility in designing the high-order undivided difference τ K , higher order of τ K is still preferred to achieve strong scale separation, which is essential for controlling the nonlinear dissipation of TENO schemes. Without adaptation of C , q and C T in the weighting strategy, increasing the order of τ K may reduce the detection-sensitivity of weak discontinuities, and consequently may decrease the robustness of the resulting TENO scheme according to extensive numerical experiments. We find that τ K of sixth-order accuracy in smooth solution regions is a suitable choice for TENO-family schemes. Deriving an analytical stability condition for the nonlinear TENO schemes is beyond the scope of this paper. By Taylor series expansion about x = xi , βk,r in Eq. (10) gives [5]
βk,r =
r +1
2
Q β2 (xi )x2 + B βk,r
=1
xr +1 + O(xr +2 ), r = 3, · · · , r dx dx x=xi
df dr f
(23)
where the first term of right hand represents the common part for all stencils and is the Gauss bracket operator. For instance, Taylor expansions of the smoothness indicators for the five- to eight-point stencils are
T (k, 5) = Q β2 x2 + Q β4 x4 + Q β6 x6 + B β5 [ T (k, 6) = Q β2 x2 + Q β4 x4 + Q β6 x6 + B β6 [
df d5 f dx dx5 df d6 f dx dx6
]x=xi x6 + O(x7 ), ]x=xi x7 + O(x8 ),
T (k, 7) = Q β2 x2 + Q β4 x4 + Q β6 x6 + Q β8 x8 + B β7 [ T (k, 8) = Q β2 x2 + Q β4 x4 + Q β6 x6 + Q β8 x8 + B β8 [
df d7 f dx dx7 df d8 f dx dx8
]x=xi x8 + O(x9 ), ]x=xi x9 + O(x10 ).
(24)
Recognizing that
1 6
2 2 df 13 d2 f [ 2 ]x=xi x4 + O(x6 ), (β1,3 + β2,3 + 4β0,3 ) = [ ]x=xi x2 + dx
12
2
dx
4
= Q β2 x + Q β4 x + O(x6 ), by canceling the common part in the Taylor expansion, a sixth-order
1 τ K = β K − (β1,3 + β2,3 + 4β0,3 ) = O(x6 ),
(25)
τ K can be constructed as
6
(26)
where β K measures the global smoothness on the K -point full stencil, for any K th-order TENO scheme (higher than fourthorder). Taking the eight-point full stencil as an example, the explicit expression of the sixth-order τ8 is
τ8 =
1 | f i +4 (75349098471 f i +4 − 1078504915264 f i +3 + 3263178215782 f i +2 62270208000 − 5401061230160 f i +1 + 5274436892970 f i − 3038037798592 f i −1
+ 956371298594 f i −2 − 127080660272 f i −3 ) + f i +3 (3944861897609 f i +3 − 24347015748304 f i +2 + 41008808432890 f i +1 − 40666174667520 f i + 23740865961334 f i −1 − 7563868580208 f i −2 + 1016165721854 f i −3 ) + f i +2 (38329064547231 f i +2 − 131672853704480 f i +1 + 132979856899250 f i − 78915800051952 f i −1 + 25505661974314 f i −2 − 3471156679072 f i −3 ) + f i +1 (115451981835025 f i +1 − 238079153652400 f i + 144094750348910 f i −1 − 47407534412640 f i −2 + 6553080547830 f i −3 ) + f i (125494539510175 f i
L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
103
− 155373333547520 f i −1 + 52241614797670 f i −2 − 7366325742800 f i −3 ) + f i −1 (49287325751121 f i −1 − 33999931981264 f i −2 + 4916835566842 f i −3 ) + f i −2 (6033767706599 f i −2 − 1799848509664 f i −3 ) + 139164877641 f i −3 f i −3 )|.
(27)
3. Unified framework for spectral property optimization In this section, the mathematic formulations of unified spectral-property optimization framework are fist presented. The dispersion and dissipation optimization of nine-point finite-difference scheme are given in detail. The spectral properties of the developed schemes are studied by the ADR analysis [22]. 3.1. The mathematical formula of unified framework Consider the one-dimensional linear advection equation
∂u ∂u +c = 0, −∞ < x < +∞, ∂t ∂x
(28)
where the advection velocity c is constant. With the discretization on a uniform mesh, e.g. x j = j x = jh, a system of ordinary differential equations is obtained as
du j dt
= −cu j , j = 0, · · · , N ,
(29)
where with linear finite-difference scheme, the first spatial derivative u j can be approximated explicitly by
u j =
r 1
h
dl u j +l ,
(30)
l=−q
and dl denotes the coefficient depending on specific scheme. Assuming that the initial condition is a monochromatic sinusoidal wave of wavelength λ and wavenumber w = 2λπ , i.e. u (x, t = 0) = e i wx , the numerical solution by Eq. (30) gives
u j (t ) = e w˜ I ct e i w ( jh−ct w˜ R / w ) ,
(31)
where the modified wave number is defined as
˜ (w) = w ˜ I =− ˜ R + iw w
r i
h
dl e ilwh =
l=−q
r 1
h
dl [sin(lwh) − i cos(lwh)].
(32)
l=−q
Compared with the analytical solution u j (t ) = e i w ( jh−ct ) , the real part of the modified wavenumber introduces extra numerical dispersion which changes the phase speed while the imaginary part induces numerical dissipation which affects the wave amplitude. With the notation of so-called reduced wavenumber ξ = wh [22], we get ξ˜ = Based on the observation that for the numerical dispersion ξ˜ R = the numerical dissipation ξ˜ I = −
r l=−q
r l=−q
r
l=−q
dl [sin(lξ ) − i cos(lξ )].
dl sin(lξ ) the term sin(lξ ) is a odd function and for
dl cos(lξ ) the term cos(lξ ) is an even function in terms of l, a family of finite-difference
schemes can be constructed with dispersion and dissipation controlled separately as
D f (x) = D dispersion f (x) + D dissipation f (x),
(33)
where
D dispersion f (x) =
n 1
x
a j ( f (x + j x) − f (x − j x))
(34)
j =1
without dissipation errors and
D dissipation f (x) =
1
x
(b0 f (x) +
n j =1
b j ( f (x + j x) + f (x − j x)))
(35)
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L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
Fig. 2. Dispersion property and phase error for various optimized sixth-order linear schemes, i.e. p = 4 in Eq. (38).
without dispersion errors. With this framework, the dispersion and dissipation property can be adjusted independently. Note that the present framework offers more flexibility than that of Sun [14] by allowing for enforcing the accuracy-order constraint separately for D dispersion f (x) and D dissipation f (x) in a simple way. In the following, the optimization of finite-difference scheme with nine points, i.e. n = 4, will be our main focus. 3.2. Optimization of the dispersion property A Taylor series expansion about j = 0, Eq. (34) gives
df D dispersion f (x) = (2a1 + 4a2 + 6a3 + 8a4 ) dx 0 G1
3 1 8 64 d f + ( a1 + a2 + 9a3 + a4 )x2 3 3 dx3 0 3 G2
1
8
81
256
4
+ ( a1 + a2 + a3 + a4 )x 15 15 60 20
d5 f dx5
0
G3
7 1 16 243 2048 d f +( a1 + a2 + a3 + a4 )x6 2520 315 280 315 dx7 0 G4 8
+ O(x ).
(36)
In order to represent the first spatial derivative, the relation 2a1 + 4a2 + 6a3 + 8a4 = 1 must be satisfied. The formal accuracy order is determined by the next non-vanishing term. For instance, if all the next three terms vanish, a linear system with four equations and four variables is formed. The solution of this linear system is unique and makes the difference operator an eighth-order accurate approximation of the first derivative. Relaxing the accuracy-order requirement, more freedom is obtained to optimize the dispersion property, e.g. for bandwidth resolution optimization according to some objective functions. Inspired by Weirs and Candler [12], an error-weighted integral function is adopted as
π E= 0
e ν (π −ξ ) (ξ˜ R − ξ )2 dξ,
(37)
L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
105
Fig. 3. Dispersion property and phase error for various optimized linear schemes compared with the standard sixth- and eighth-order central schemes. On the right, the long dash-dotted lines give the error bound of ε = 1.5%.
Fig. 4. Left: dissipation property; right: dispersion-dissipation relation ζ = dispersion part is determined with D dispersion f (x) of p = 4.
dξ˜ R −1+10−3 dξ −ξ˜ I +10−3
for various optimized linear schemes with q = 4, in which the
where the positive parameter ν is designed to control the relative importance of low wavenumber and high wavenumber dispersion errors. The minimization of Eq. (37) gives the condition ∂∂aE = 0. The optimization of dispersion errors with order i constraint thus can be generalized as
⎧ ⎨ G 1 = 1, G i = 0, 1 < i < p , ⎩ ∂ E = 0, p i 4, ∂ ai
(38)
where the accuracy order of the generated scheme is 2( p − 1), and the solution is unique without free parameters. As shown on the left of Fig. 2, a larger parameter ν results in a sixth-order linear scheme with improved dispersion property as the integral of weighted error is smaller. However, the dispersion property degenerates quickly in the low wavenumber region of interest. Weirs and Candler [12] suggest that the phase discrepancy
ε=
ξ˜ R ξ
− 1 should be less than
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L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
Fig. 5. Left: dissipation property; right: dispersion-dissipation relation ζ = dispersion part is determined with D dispersion f (x) of p = 3.
dξ˜ R −1+10−3 dξ −ξ˜ I +10−3
for various optimized linear schemes with q = 3, in which the
Fig. 6. g 1 (x) and its first derivative
∂ g 1 ( x) ∂x .
1.5% at the wavenumber of interest for the optimization procedure. According to the right of Fig. 2, the parameter ν = 7 is adopted in the following without tuning. Fig. 3 shows the dispersion properties of optimized linear schemes with different order requirements. Compared with the standard sixth-order and eighth-order central schemes, the dispersion property is significantly improved for intermediate wavenumber range 1 ≤ ξ ≤ 2.2. While better spectral property is obtained with larger accuracy-order reduction, the cost-effectiveness decreases remarkably. As recommended by Lele [23], the resolved bandwidth is defined as ξ with which the phase error first exceeds the threshold ε = 1.5%. From the right of Fig. 3, it can be seen that the resolved bandwidth gradually improves from 1.17, 1.41 to 1.75, 1.89, 1.95 corresponding to the standard sixth-, eighth-order central scheme and the optimized sixth-, fourth-, second-order scheme, respectively. The dispersion properties of the optimized sixth-, fourthand second-order scheme are comparable to that of the sixth-order compact scheme [23]. 3.3. The optimization of dissipation property By Taylor series expansion about j = 0, Eq. (35) gives
D dissipation f (x) = (b0 + 2b1 + 2b2 + 2b3 + 2b4 )
G0
1
x
( f )0
L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
Fig. 7. Left: dissipation property; right: dispersion-dissipation relation ζ =
dξ˜ R −1+10−3 dξ −ξ˜ I +10−3
dispersion part is determined with D dispersion f (x) of p = 2.
107
for various optimized linear schemes with q = 2, in which the
2 d f + (b1 + 4b2 + 9b3 + 16b4 )x dx2 0 G1
1
4
27
64
3
+ ( b1 + b2 + b3 + b4 )x 3 3 12 4
d4 f dx4
0
G2
6 1 8 81 512 d f +( b1 + b2 + b3 + b4 )x5 45 40 45 dx6 0 360 G3
8 1 4 729 1024 d f +( b1 + b2 + b3 + b4 )x7 20160 315 2240 315 dx8 0 G4
9
+ O(x ).
(39)
Similarly, the accuracy order is determined by the non-vanishing leading-order term. If the desired truncation order is O (x9 ), the solution of the linear system Gi = 0, 0 ≤ i ≤ 4 leads to b i = 0, 0 ≤ i ≤ 4. Therefore, the dissipation error is zero for all wavenumbers. Since the dispersion error does not vanish from intermediate to high wavenumbers for any explicit finite-difference scheme, adequate numerical dissipation is necessary for the sake of numerical stability. Hu et al. [24] propose an approximate dispersion-dissipation relation for finite-difference schemes for estimating the numerical dissipation necessary to damp spurious high wave-number errors as
ζ=
dξ˜R d ξ − 1 −ξ˜ I
≈ O(10).
(40)
Given this relation, the objective numerical dissipation for certain wavenumber can be calculated independently without affecting the dispersion property. By dropping the accuracy order, we gain the freedom to control the dissipation amount and the dissipation-wavenumber relation. With q ≤ 4, a set of linear equations can be constructed as
⎧ 0 i < q, ⎨ G i = 0, ξ˜ I (ξi ) = c i , q i 3, ⎩ b0 i = 4,
(41)
where 2q − 1 order accuracy is retained for the linear scheme and the free parameter b0 is determined by enforcing a monotonicity condition for the dissipation function. When q = 4, the second equation in Eq. (41) vanishes correspondingly.
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L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
Fig. 8. Dispersion property, dissipation property and dispersion-dissipation relation for TENO8 (left) and TENO8-opt (right) scheme with different cutoff C T . On the bottom panel, ζ =
dξ˜ R −1+10−3 dξ −ξ˜ I +10−3
.
L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
109
Fig. 9. Convergence of the L ∞ error from the TENO8 and TENO8-opt scheme.
Fig. 10. Convergence of the L ∞ error from the TENO8 and TENO8-opt scheme for the test function with second-order critical points.
First consider q = 4. The imaginary part of the modified wavenumber can be rewritten as
ξ˜ I (ξ ) = −
r
dl cos(lξ ),
l=−q
= −b0
8(cos(ξ ) − 1)4 35
(42)
.
It is obvious that the numerical dissipation is strictly non-negative for all wavenumbers as long as b0 ≥ 0. Assuming that ∂ ξ˜
32b (x−1)3
x = cos(ξ ) and −1 ≤ x ≤ 1 for 0 ≤ ξ ≤ π , ∂ xI = − 035 ≥ 0 with the condition b0 ≥ 0, thus ξ˜ I (ξ ) is a monotonically non-increasing function in terms of ξ since x is a monotonically decreasing function in terms of ξ for 0 ≤ ξ ≤ π . Fig. 4 gives the dissipation property and dispersion-dissipation relation for various linear schemes. By adjusting the free parameter b0 , the absolute value of numerical dissipation can be tailored and the approximate relation Eq. (40) can be fulfilled. For the case q = 3, an additional option to control the numerical dissipation at a specific wavenumber is available. Typically the maximum wavenumber π is adopted as control point enforcing the condition Eq. (40), e.g. ξ˜ I (π ) = −0.24 according to Eq. (40). Since ξ˜ I (0) = b0 + 2b1 + 2b2 + 2b3 + 2b4 = 0 is naturally satisfied by accuracy-order constraint and ξ˜ I (π ) < 0, the condition for positive dissipation at all wavenumbers is equivalent to the requirement of monotonicity of the imaginary part of the modified wavenumber. After some simplification, it is
110
L. Fu et al. / Journal of Computational Physics 349 (2017) 97–121
Fig. 11. Convergence of the L ∞ error from the TENO8 and TENO8-opt scheme for 1D inviscid nonlinear Burger’s equation.
ξ˜ I (ξ ) =
(cos(ξ ) − 1)3 (160b0 cos(ξ ) − 12 cos(ξ ) + 160b0 − 9) 100
(43)
.
With the notation x = cos(ξ ) and −1 ≤ x ≤ 1 for 0 ≤ ξ ≤ π ,
(x − 1)2 (640b0 x − 48x + 320b0 − 15) ∂ ξ˜ I = . ∂x 100
(44) ∂ ξ˜
Considering that g (x) = 640b0 x − 48x + 320b0 − 15, the monotonicity condition, i.e. ∂ x ≥ 0, is equivalent to
g (1) ≥ 0, g (−1) ≥ 0.
(45)
Therefore, the free parameter is constrained by 0.065625 ≤ b0 ≤ 0.103125. As demonstrated in Fig. 5, the dissipation at ξ = π remains unchanged and an adjustment of parameter b0 controls the dissipation function in wave space. Moreover, anti-dissipation can be produced in the intermediate wavenumber region if the choice of b0 violates the monotonicity condition. Another control point can be used to constrain numerical dissipation further if q = 2, e.g. ξ˜ I ( π2 ) = −0.02 and
ξ˜ I (π ) = −0.28 according to Eq. (40). The imaginary part of the modified wavenumber can be derived as
ξ˜ I (ξ ) =
(cos(ξ ) − 1)2 (800b0 cos2 (ξ ) − 64cos2 (ξ ) + 800b0 cos(ξ ) − 59 cos(ξ ) − 2) 100
.
(46)
Similar to the case q = 3, the monotonicity of function ξ˜ I (ξ ) can ensure non-negative dissipation for all wavenumbers. The derivative of ξ˜ I (ξ ) gives
(x − 1)(3200b0 x2 − 256x2 + 800b0 x − 49x − 800b0 + 55) ∂ ξ˜ I = , ∂x 100 where x = cos(ξ ) and −1 ≤ x ≤ 1 for 0 ≤ ξ ≤ π . Monotonicity requires that g 0 (x) ≤ 0, for − 1 ≤ x ≤ 1,
(47)
(48)
where g 0 (x) = (3200b0 x2 − 256x2 + 800b0 x − 49x − 800b0 + 55). Defining
g 1 (x) =
256x2 + 49x − 55 3200x2 + 800x − 800
(49)
,
the condition becomes
b0 ≥ g 1 (x), b0 < g 1 (x),
if −
√
17+1 8
