Schemes For Hyperbolic Conservation Laws Research Articles

Upwind bicompact schemes for hyperbolic conservation laws

Upwind bicompact schemes for hyperbolic conservation laws

High-order oscillation-eliminating Hermite WENO method for hyperbolic conservation laws

This paper proposes high-order accurate, oscillation-eliminating, Hermite weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for hyperbolic conservation laws, motivated by the oscillation-eliminating (OE) discontinuous Galerkin schemes recently proposed in [M. Peng, Z. Sun, and K. Wu, 2024 [30]]. The OE-HWENO schemes incorporate an OE procedure after each Runge–Kutta stage, by dampening the first-order moments of the HWENO solution to suppress spurious oscillations without any problem-dependent parameter. The OE procedure acts as a moment filter and is derived from the solution operator of a novel damping equation, which is exactly solved without any discretization. Thanks to this distinctive feature, the OE-HWENO method remains stable with a normal CFL number, even for strong shocks resulting in highly stiff damping terms. To ensure the essentially non-oscillatory property of the OE-HWENO method across problems with varying scales and wave speeds, we design a scale-invariant and evolution-invariant damping equation and propose a generic dimensionless transformation for HWENO reconstruction. The OE-HWENO method offers several notable advantages over existing HWENO methods. First, the OE procedure is highly efficient and straightforward to implement, requiring only simple multiplication of first-order moments by a damping factor. Furthermore, we rigorously prove that the OE procedure maintains the high-order accuracy and local compactness of the original HWENO schemes and demonstrate that it does not compromise the spectral properties via the approximate dispersion relation for smooth solutions. Notably, the proposed OE procedure is non-intrusive, enabling seamless integration as an independent module into existing HWENO codes. Finally, we rigorously analyze the bound-preserving (BP) property of the OE-HWENO method using the optimal cell average decomposition approach [S. Cui, S. Ding, and K. Wu, 2024 [8]], which relaxes the theoretical BP constraint for time step-size and reduces the number of decomposition points, thereby further enhancing efficiency. Extensive benchmarks validate the accuracy, efficiency, high resolution, and robustness of the OE-HWENO method.

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess both scale-invariant and evolution-invariant properties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate for nonlinear DG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure is nonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques. Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method.

Sixth-order perturbed WENO interpolation-based AWENO and WCNS-E schemes for hyperbolic conservation laws

The weighted essentially non-oscillatory (WENO) interpolation-based schemes (the alternative WENO (AWENO) scheme and the explicit weighted compact nonlinear (WCNS-E) scheme) are limited to the fifth-order despite the sixth-order Taylor expansion of the numerical flux used. This order reduction is due to the fifth-order accuracy inherent in the WENO interpolation. We investigate the perturbed WENO interpolation with a free parameter and affine-invariant WENO weights to recover sixth-order accuracy in smooth regions. A cutoff function of the free parameter with a threshold, determined by approximate dispersion relation analysis, is applied to enhance the ENO property. The proposed schemes perform better in accuracy, dissipation, resolution, shock-capturing, and efficiency in 1D and 2D benchmark problems.

Upwind Bicompact Schemes for Hyperbolic Conservation Laws

Upwind Bicompact Schemes for Hyperbolic Conservation Laws

A new global smoothness indicator of fifth order weighted non-oscillatory scheme for hyperbolic conservation laws

A new global smoothness indicator of fifth order weighted non-oscillatory scheme for hyperbolic conservation laws

A third‐order entropy condition scheme for hyperbolic conservation laws

AbstractFollowing the solution formula method given in Dong et al. (High order discontinuities decomposition entropy condition schemes for Euler equations. CFD J. 2002;10(4): 448–457), this article studies a type of one‐step fully‐discrete scheme, and constructs a third‐order scheme which is written into a compact form via a new limiter. The highlights of this study and advantages of new third‐order scheme are as follows: ① We proposed a very simple new methodology of constructing one‐step, consistent high‐order and non‐oscillation schemes that do not rely on Runge–Kutta method; ② We systematically studied new scheme's theoretical problems about entropy conditions, error analysis, and non‐oscillation conditions; ③ The new scheme achieves exact solution in linear cases and performing better in nonlinear cases when CFL → 1; ④ The new scheme is third order but high resolution with excellent shock‐capturing capacity which is comparable to fifth order WENO scheme; ⑤ CPU time of new scheme is only a quarter of WENO5 + RK3 under same computing condition; ⑥ For engineering applications, the new scheme is extended to multi‐dimensional Euler equations under curvilinear coordinates. Numerical experiments contain 1D scalar equation, 1D,2D,3D Euler equations. Accuracy tests are carried out using 1D linear scalar equation, 1D Burgers equation and 2D Euler equations and two sonic point tests are carried out to show the effect of entropy condition linearization. All tests are compared with results of WENO5 and finally indicate EC3 is cheaper in computational expense.

New Finite Volume Mapped Unequal-Sized WENO Scheme for Hyperbolic Conservation Laws

New Finite Volume Mapped Unequal-Sized WENO Scheme for Hyperbolic Conservation Laws

High-order adaptive multiresolution wavelet upwind schemes for hyperbolic conservation laws

We present a novel system of high-order adaptive multiresolution wavelet collocation upwind schemes, implemented in a meshfree framework, for solving hyperbolic conservation laws. Our approach constructs asymmetrical wavelet bases with interpolation properties to achieve the upwind property and address nonlinearities in hyperbolic problems. An adaptive algorithm based on multiresolution analysis in wavelet theory is designed to capture moving shock waves and identify new localized steep regions. To suppress the Gibbs phenomenon, we propose an integration average reconstruction method based on the Lebesgue differentiation theorem. These numerical techniques enable the wavelet collocation upwind scheme to provide a general framework for devising satisfactory adaptive wavelet upwind methods with high-order accuracy. We perform several benchmark tests for 1D hyperbolic problems to verify the accuracy and efficiency of the proposed wavelet schemes. Notably, our wavelet collocation upwind schemes successfully solve the two interacting blast waves problem when discretizing the spatial derivatives in the physical space directly. This is in contrast to classical high-order schemes that usually involve transforming variables back and forth between physical and characteristic spaces with many local projections. Our results indicate the robustness, stability, and efficiency of our proposed schemes for strong shock wave interaction problems.

A new type of non-polynomial based TENO scheme for hyperbolic conservation laws

For classical WENO/TENO reconstructions, the high-order polynomial interpolation is one of the main building-blocks for constructing the numerical fluxes. However, due to the inherent characteristics of polynomials, the polynomial interpolation is not suitable to represent the flow scale with sharp transitions such as high-wavenumber fluctuations or discontinuities. In this work, based on a new candidate stencil arrangement with explicit detection of non-smooth regions, a new shock-capturing framework is proposed by combining the infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with the jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme not only achieves the desired high-order accuracy for smooth flow scales but also resolves the genuine discontinuities with a sub-cell resolution. Moreover, with the observation that the built-in parameter has a significant impact on the basic shape of the Gaussian RBF kernel function, a nonlinear adaptation strategy for the shape parameter is proposed to incorporate the local features of the flow field and further minimize the numerical dissipation errors for smooth flow scales. In order to demonstrate the robust shock-capturing and high wave-resolution properties of the proposed non-polynomial based scheme, a set of challenging benchmark cases with a wide range of Mach numbers is simulated without the necessity of parameter tuning case by case.

An Improved WENO-Z Scheme for Hyperbolic Conservation Laws with New Global Smoothness Indicator

The fifth-order WENO-Z scheme proposed by Borges et al., using a linear combination of low-order smoothness indicators, is designed to provide a low numerical dissipation to solve hyperbolic conservation laws, while the power q in the framework of WENO-Z plays a key role in its performance. In this paper, a novel global smoothness indicator with fifth-order accuracy, which is based on several lower-order smoothness indicators on two-point sub-stencils, is presented, and a new lower-dissipation WENO-Z scheme (WENO-NZ) is developed. The spectral properties of the WENO-NZ scheme are studied through the ADR method and show that this new scheme can exhibit better spectral results than WENO-Z no matter what the power value is. Accuracy tests confirm that the accuracy of WENO-Z with q = 1 would degrade to the fourth order at first-order critical points, while WENO-NZ can recover the optimal fifth-order convergence. Furthermore, numerical experiments with one- and two-dimensional benchmark problems demonstrate that the proposed WENO-NZ scheme can efficiently decrease the numerical dissipation and has a higher resolution compared to the WENO-Z scheme.

WENO smoothness indicator based troubled‐cell indicator for hyperbolic conservation laws

SummaryHybrid algorithms are an efficient and popular choice for computing the solutions of hyperbolic conservation laws. In general, hybrid algorithms involve low‐cost high‐order accurate schemes in smooth regions and non‐oscillatory shock‐capturing schemes in the vicinity of discontinuities. Troubled‐cell indicators which measure the smoothness of the solution play a significant role in the efficiency of hybrid algorithms. This article proposes a new troubled‐cell indicator utilising the smoothness indicators of WENO schemes for hyperbolic conservation laws. The proposed troubled‐cell indicators are simple, efficient, effective, and are used to construct three new adaptive WENO algorithms of high‐order accuracy. The hybrid algorithms developed are independent of the order and type of the WENO reconstruction. For demonstration, we have considered the fifth and seventh order WENO‐Z reconstruction. The first two algorithms have comparable accuracy and resolution of the solution across discontinuities to that of the WENO‐Z scheme but at a less computational cost. The third algorithm ensures the convergence of the proposed scheme to the correct entropy solution when applied to a hyperbolic conservation law with non‐convex flux for which the WENO schemes fail. We have performed several 1D and 2D numerical experiments to demonstrate the efficiency of the proposed algorithms and their performance compared with the WENO‐Z schemes. The proposed algorithms are efficient and take 30%–75% less computational time than the WENO‐Z schemes while retaining the advantages of WENO‐Z schemes.

On the Convergence of α Schemes with Source Terms for Scalar Convex Conservation Laws

In this study, we use an extension of Yang’s convergence criterion [N. Jiang, On the wavewise entropy inequality for high-resolution schemes with source terms II: the fully discrete case] to show the entropy convergence of a class of fully discrete α schemes, now with source terms, for non-homogeneous scalar convex conservation laws in the one-dimensional case. The homogeneous counterparts (HCPs) of these schemes were constructed by S. Osher and S. Chakravarthy in the mid-1980s [A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws (1985), Very High Order Accurate TVD Schemes (1986)], and the entropy convergence of these methods, when m=2, was settled by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws II: Fully-Discrete]. For semi-discrete α schemes, with or without source terms, the entropy convergence of these schemes was previously established (for m=2) by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws I: Semi-Discrete Case].

A new high order hybrid WENO scheme for hyperbolic conservation laws

AbstractThis article proposes an improved hybrid weighted essentially non‐oscillatory (WENO) scheme based on the third‐ and fifth‐order finite‐difference modified WENO (MWENO) schemes developed by Zhu et al. in (SIAM J. Sci. Comput. 39 (2017), A1089–A1113.) for solving hyperbolic conservation laws. The MWENO schemes give a guideline on whether to use the WENO scheme or the linear upwind scheme. Unfortunately, because there is no explicit formula for computing the roots of algebraic polynomials of order four or higher, it is difficult to generalize this criterion to higher order cases. Therefore, this article proposes a simple criterion for constructing a series of seventh‐, ninth‐, and higher‐order hybrid WENO schemes, and then designs a class of improved smooth indicator WENO (WENO‐MS) schemes. Compared with the classical WENO schemes, the main advantages of the WENO‐MS schemes are their robustness and efficiency. And these WENO‐MS schemes are more efficient, have better resolution, and can solve many extreme problems without any additional techniques. Furthermore, a simplification criterion is proposed to further improve the computational efficiency of the WENO‐MS schemes, and these simple WENO‐MS schemes are abbreviated as WENO‐SMS schemes in this article. Extensive numerical results demonstrate the good performance of the WENO‐MS schemes and the WENO‐SMS schemes.

A Seventh-order Perturbational Weighted Essentially Non-oscillatory Scheme for Hyperbolic Conservation Laws

A Seventh-order Perturbational Weighted Essentially Non-oscillatory Scheme for Hyperbolic Conservation Laws

A Flux Reconstruction Stochastic Galerkin Scheme for Hyperbolic Conservation Laws

The study of uncertainty propagation poses a great challenge to design high fidelity numerical methods. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. High-order numerical approximation is adopted simultaneously in physical and random space, i.e., the modal representation of solutions is based on an orthogonal polynomial basis and the nodal representation is based on solution collocation points. Therefore, the numerical behaviors of the scheme in the (physical-random) phase space can be designed and understood uniformly. A family of filters is developed in multi-dimensional cases to mitigate the Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture the stochastic flow evolution where resolved and unresolved regions coexist. Numerical experiments including a wave propagation, a Burgers’ shock, a one-dimensional Riemann problem, and a two-dimensional shock-vortex interaction problem are presented to validate the current scheme. The order of convergence of the high-order scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license (Xiao et al. in FRSG: stochastic Galerkin method with flux reconstruction. https://github.com/CSMMLab/FRSG, (2021). https://doi.org/10.5281/zenodo.5588317).

Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws

The numerical solution of hyperbolic conservation laws requires algorithms with upwind characteristics. Conventional methods such as the numerical difference method can realize this characteristic by constructing special distributions of nodes. However, there are still no reports on how to construct algorithms with upwind characteristics through wavelet theory. To solve this problem, a system of high-order and stable wavelet collocation upwind schemes was successfully proposed by constructing interpolation wavelets with specific symmetry and smoothness. The effects of the characteristics of the scaling functions on the schemes were explored based on numerical tests and Fourier analysis. The numerical results revealed that the stability of the constructed scheme is affected by the smoothness order, N, and the asymmetry of the scaling function. The dissipation analysis suggested that schemes with N ∈ even have negative dissipation coefficients, leading to unstable behaviors. Only scaling functions with N ∈ odd and a bias magnitude of 1 can be used to construct stable upwind schemes due to the non-negative dissipation coefficients. Typical numerical examples verified the effectiveness of the proposed method, which is proved to have high accuracy and efficiency in solving high-speed flow problems with multi-scale smooth structures and discontinuities.

A Scale-Invariant Fifth Order WCNS Scheme for Hyperbolic Conservation Laws

A Scale-Invariant Fifth Order WCNS Scheme for Hyperbolic Conservation Laws

The Formulation of Finite Difference RBFWENO Schemes for Hyperbolic Conservation Laws: An Alternative Technique

The Formulation of Finite Difference RBFWENO Schemes for Hyperbolic Conservation Laws: An Alternative Technique

A third-order weighted nonlinear scheme for hyperbolic conservation laws with inverse Lax-Wendroff boundary treatment

Cartesian grids are often used in applications due to the simplicity of grid generation and the efficiency of discretization algorithms. However, for problems with curved boundaries, grid points are often away from the boundaries, leading to the issue of imposing boundary conditions. Recently, it was shown that the inverse Lax-Wendroff (ILW) boundary treatment is very useful for addressing this issue. While most of the investigations focus on the WENO schemes for hyperbolic conservation laws, we present in this paper a new third-order weighted nonlinear scheme, which is based on the framework of weighted compact nonlinear schemes. We show that the scheme is applicable directly on Cartesian grids by using the ILW boundary treatment. To demonstrate the linear stability of the scheme, both the method of eigenvalue analysis and the Gustafsson-Kreiss-Sundström (GKS) theory are employed and analyzed in details. Some numerical tests are also performed to show the accuracy and validity of the proposed method.