Table of Content

20 March 2023, Volume 5 Issue 1

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PREFACE

Preface to the Focused Issue on WENO Schemes

Sigal Gottlieb, Jan S. Hesthaven, Jianxian Qiu, Chi-Wang Shu, Qiang Zhang, Yong-Tao Zhang

2023, 5(1):  1-2.  doi:10.1007/s42967-022-00196-z

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ORIGINAL PAPERS

A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws

Kunlei Zhao, Yulong Du, Li Yuan

2023, 5(1):  3-30.  doi:10.1007/s42967-020-00112-3

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In this paper, we develop a new sixth-order WENO scheme by adopting a convex combination of a sixth-order global reconstruction and four low-order local reconstructions. Unlike the classical WENO schemes, the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one. Further, a very simple smoothness indicator for the global stencil is proposed. The new scheme can achieve sixthorder accuracy in smooth regions. Numerical tests in some one- and two-dimensional benchmark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme, and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.

High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations

Jie Du, Yang Yang

2023, 5(1):  31-63.  doi:10.1007/s42967-020-00117-y

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In this paper, we apply high-order finite difference (FD) schemes for multispecies and multireaction detonations (MMD). In MMD, the density and pressure are positive and the mass fraction of the ith species in the chemical reaction, say z_i, is between 0 and 1, with Σz_i = 1. Due to the lack of maximum-principle, most of the previous bound-preserving technique cannot be applied directly. To preserve those bounds, we will use the positivity-preserving technique to all the z_i'is and enforce Σz_i = 1 by constructing conservative schemes, thanks to conservative time integrations and consistent numerical fluxes in the system. Moreover, detonation is an extreme singular mode of flame propagation in premixed gas, and the model contains a significant stiff source. It is well known that for hyperbolic equations with stiff source, the transition points in the numerical approximations near the shocks may trigger spurious shock speed, leading to wrong shock position. Intuitively, the high-order weighted essentially non-oscillatory (WENO) scheme, which can suppress oscillations near the discontinuities, would be a good choice for spatial discretization. However, with the nonlinear weights, the numerical fluxes are no longer “consistent”, leading to nonconservative numerical schemes and the bound-preserving technique does not work. Numerical experiments demonstrate that, without further numerical techniques such as subcell resolutions, the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.

New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy

Jun Zhu, Jianxian Qiu

2023, 5(1):  64-96.  doi:10.1007/s42967-021-00122-9

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In this paper, a new type of finite difference mapped weighted essentially non-oscillatory (MWENO) schemes with unequal-sized stencils, such as the seventh-order and ninthorder versions, is constructed for solving hyperbolic conservation laws. For the purpose of designing increasingly high-order finite difference WENO schemes, the equal-sized stencils are becoming more and more wider. The more we use wider candidate stencils, the bigger the probability of discontinuities lies in all stencils. Therefore, one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point stencils. By the usage of this new methodology in high-order spatial reconstruction procedure, we get different degree polynomials defined on these unequal-sized stencils, and calculate the linear weights, smoothness indicators, and nonlinear weights as specified in Jiang and Shu (J. Comput. Phys. 126: 202228, 1996). Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions, another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights, so as to keep the optimal order of accuracy in smooth regions. These new MWENO schemes can also be applied to compute some extreme examples, such as the double rarefaction wave problem, the Sedov blast wave problem, and the Leblanc problem with a normal CFL number. Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.

A GPU-Accelerated Mixed-Precision WENO Method for Extremal Black Hole and Gravitational Wave Physics Computations

Scott E. Field, Sigal Gottlieb, Zachary J. Grant, Leah F. Isherwood, Gaurav Khanna

2023, 5(1):  97-115.  doi:10.1007/s42967-021-00129-2

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We develop and use a novel mixed-precision weighted essentially non-oscillatory (WENO) method for solving the Teukolsky equation, which arises when modeling perturbations of Kerr black holes. We show that WENO methods outperform higher-order finite-difference methods, standard in the discretization of the Teukolsky equation, due to the need to add dissipation for stability purposes in the latter. In particular, as the WENO scheme uses no additional dissipation, it is well suited for scenarios requiring long-time evolution such as the study of price tails and gravitational wave emission from extreme mass ratio binaries. In the mixed-precision approach, the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of ≈ 3.3. In addition, we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations. Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics. We apply our solver to study the behavior of the Aretakis charge—a conserved quantity, that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.

High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation

Andrew Christlieb, Matthew Link, Hyoseon Yang, Ruimeng Chang

2023, 5(1):  116-142.  doi:10.1007/s42967-021-00150-5

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In this paper, we present a semi-Lagrangian (SL) method based on a non-polynomial function space for solving the Vlasov equation. We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems. To address issues that arise in phase space models of plasma problems, we develop a weighted essentially non-oscillatory (WENO) scheme using trigonometric polynomials. In particular, the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities. Moreover, to obtain a high-order of accuracy in not only space but also time, it is proposed to apply a high-order splitting scheme in time. We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system. Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions. A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method. In 6D, this would represent a signifcant savings.

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

M. Semplice, E. Travaglia, G. Puppo

2023, 5(1):  143-169.  doi:10.1007/s42967-021-00151-4

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We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252–270. https:// doi. org/ 10. 1016/j. amc. 2017. 12. 041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells.

A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

Linjin Li, Jingmei Qiu, Giovanni Russo

2023, 5(1):  170-198.  doi:10.1007/s42967-021-00156-z

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In this paper, we propose a new conservative high-order semi-Lagrangian finite difference (SLFD) method to solve linear advection equation and the nonlinear Vlasov and BGK models. The finite difference scheme has better computational flexibility by working with point values, especially when working with high-dimensional problems in an operator splitting setting. The reconstruction procedure in the proposed SLFD scheme is motivated from the SL finite volume scheme. In particular, we define a new sliding average function, whose cell averages agree with point values of the underlying function. By developing the SL finite volume scheme for the sliding average function, we derive the proposed SLFD scheme, which is high-order accurate, mass conservative and unconditionally stable for linear problems. The performance of the scheme is showcased by linear transport applications, as well as the nonlinear Vlasov-Poisson and BGK models. Furthermore, we apply the Fourier stability analysis to a fully discrete SLFD scheme coupled with diagonally implicit Runge-Kutta (DIRK) method when applied to a stiff two-velocity hyperbolic relaxation system. Numerical stability and asymptotic accuracy properties of DIRK methods are discussed in theoretical and computational aspects.

A New Hybrid WENO Scheme with the High-Frequency Region for Hyperbolic Conservation Laws

Yifei Wan, Yinhua Xia

2023, 5(1):  199-234.  doi:10.1007/s42967-021-00153-2

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In this paper, a new kind of hybrid method based on the weighted essentially non-oscillatory (WENO) type reconstruction is proposed to solve hyperbolic conservation laws. Comparing the WENO schemes with/without hybridization, the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region; meanwhile, the essentially oscillation-free solution could also be obtained. By adapting the original smoothness indicator in the WENO reconstruction, the stencil is distinguished into three types: smooth, non-smooth, and high-frequency region. In the smooth region, the linear reconstruction is used and the non-smooth region with the WENO reconstruction. In the high-frequency region, the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor, which could capture high-frequency wave efficiently. Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.

Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

Dinshaw S. Balsara, Roger Käppeli, Walter Boscheri, Michael Dumbser

2023, 5(1):  235-294.  doi:10.1007/s42967-021-00160-3

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Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called firstorder reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENOlike schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

Fifth-Order A-WENO Schemes Based on the Adaptive Diffusion Central-Upwind Rankine-Hugoniot Fluxes

Bao-Shan Wang, Wai Sun Don, Alexander Kurganov, Yongle Liu

2023, 5(1):  295-314.  doi:10.1007/s42967-021-00161-2

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We construct new fifth-order alternative WENO (A-WENO) schemes for the Euler equations of gas dynamics. The new scheme is based on a new adaptive diffusion centralupwind Rankine-Hugoniot (CURH) numerical flux. The CURH numerical fluxes have been recently proposed in [Garg et al. J Comput Phys 428, 2021] in the context of secondorder semi-discrete finite-volume methods. The proposed adaptive diffusion CURH flux contains a smaller amount of numerical dissipation compared with the adaptive diffusion central numerical flux, which was also developed with the help of the discrete RankineHugoniot conditions and used in the fifth-order A-WENO scheme recently introduced in [Wang et al. SIAM J Sci Comput 42, 2020]. As in that work, we here use the fifth-order characteristic-wise WENO-Z interpolations to evaluate the fifth-order point values required by the numerical fluxes. The resulting one- and two-dimensional schemes are tested on a number of numerical examples, which clearly demonstrate that the new schemes outperform the existing fifth-order A-WENO schemes without compromising the robustness.

A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

Xucheng Meng, Yaguang Gu, Guanghui Hu

2023, 5(1):  315-342.  doi:10.1007/s42967-021-00163-0

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In Li and Ren (Int. J. Numer. Methods Fluids 70: 742–763, 2012), a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain, in which the high-order numerical accuracy and the oscillations-free property can be achieved. In this paper, the method is extended to solve steady state problems imposed in a curved physical domain. The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations, and a geometrical multigrid method to solve the derived linear system. To achieve high-order non-oscillatory numerical solutions, the classical k-exact reconstruction with k = 3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables. The non-uniform rational B-splines (NURBS) curve is used to provide an exact or a high-order representation of the curved wall boundary. Furthermore, an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state. A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

Quinpi: Integrating Conservation Laws with CWENO Implicit Methods

G. Puppo, M. Semplice, G. Visconti

2023, 5(1):  343-369.  doi:10.1007/s42967-021-00171-0

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Many interesting applications of hyperbolic systems of equations are stiff, and require the time step to satisfy restrictive stability conditions. One way to avoid small time steps is to use implicit time integration. Implicit integration is quite straightforward for first-order schemes. High order schemes instead also need to control spurious oscillations, which requires limiting in space and time also in the linear case. We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor, which is used both to set up the nonlinear weights of a standard high order space reconstruction, and to achieve limiting in time. In this preliminary work, we concentrate on the case of a third-order scheme, based on diagonally implicit Runge Kutta (DIRK) integration in time and central weighted essentially non-oscillatory (CWENO) reconstruction in space. The numerical tests involve linear and nonlinear scalar conservation laws.

A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem: Application to the 1D Euler Equations

R. Abgrall

2023, 5(1):  370-402.  doi:10.1007/s42967-021-00175-w

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We show how to combine in a natural way (i.e., without any test nor switch) the conservative and non-conservative formulations of an hyperbolic system that has a conservative form. This is inspired from two different classes of schemes: the residual distribution one (Abgrall in Commun Appl Math Comput 2(3): 341–368, 2020), and the active flux formulations (Eyman and Roe in 49th AIAA Aerospace Science Meeting, 2011; Eyman in active flux. PhD thesis, University of Michigan, 2013; Helzel et al. in J Sci Comput 80(3): 35–61, 2019; Barsukow in J Sci Comput 86(1): paper No. 3, 34, 2021; Roe in J Sci Comput 73: 1094–1114, 2017). The solution is globally continuous, and as in the active flux method, described by a combination of point values and average values. Unlike the “classical” active flux methods, the meaning of the point-wise and cell average degrees of freedom is different, and hence follow different forms of PDEs; it is a conservative version of the cell average, and a possibly non-conservative one for the points. This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem. We also develop a method to perform nonlinear stability. We illustrate the behaviour on several benchmarks, some quite challenging.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Liang Li, Jun Zhu, Chi-Wang Shu, Yong-Tao Zhang

2023, 5(1):  403-427.  doi:10.1007/s42967-021-00179-6

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Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show highorder accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

Efficient WENO-Based Prolongation Strategies for Divergence-Preserving Vector Fields

Dinshaw S. Balsara, Saurav Samantaray, Sethupathy Subramanian

2023, 5(1):  428-484.  doi:10.1007/s42967-021-00182-x

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Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh. For scalar variables, suitably high-order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, such as computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces, where the vector field components are collocated. This approach is almost divergence constraint-preserving, therefore, we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods, where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields, where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.

High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

Zepeng Liu, Yan Jiang, Mengping Zhang, Qingyuan Liu

2023, 5(1):  485-528.  doi:10.1007/s42967-021-00183-w

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A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory (WENO) scheme. The exact C-property is investigated, and comparison with the standard finite difference WENO scheme is made. Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems. The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems, indicating smaller errors compared with the Lax-Friedrichs solver. In addition, we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.