Schemes For Hyperbolic Conservation Laws Research Articles

A new adaptive weighted essentially non-oscillatory WENO-θ scheme for hyperbolic conservation laws

A new adaptive weighted essentially non-oscillatory WENO- θ scheme in the context of finite difference is proposed. Depending on the smoothness of the large stencil used in the reconstruction of the numerical flux, a parameter θ is set adaptively to switch the scheme between a 5th-order upwind and 6th-order central discretization. A new indicator τ θ measuring the smoothness of the large stencil is chosen among two candidates which are devised based on the possible highest-order variations of the reconstruction polynomials in L 2 sense. In addition, a new set of smoothness indicators β ̃ k of the substencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around the point x j . Numerical results show that the new scheme outperforms other comparing 6th-order WENO schemes in terms of improving the resolution at critical regions of nonsmooth problems as well as maintaining symmetry in the solutions.

A New Type of Finite Volume WENO Schemes for Hyperbolic Conservation Laws

A new type of finite difference weighted essentially non-oscillatory (WENO) schemes for hyperbolic conservation laws was designed in Zhu and Qiu (J Comput Phys 318:110–121, 2016), in this continuing paper, we extend such methods to finite volume version in multi-dimensions. There are two major advantages of the new WENO schemes superior to the classical finite volume WENO schemes (Shu, in: Quarteroni (ed) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, CIME subseries, Springer, Berlin, 1998), the first is the associated linear weights can be any positive numbers with only requirement that their summation equals one, and the second is their simplicity and easy extension to multi-dimensions in engineering applications. The new WENO reconstruction is a convex combination of a fourth degree polynomial with two linear polynomials defined on unequal size spatial stencils in a traditional WENO fashion. These new fifth order WENO schemes use the same number of cell average information as the classical fifth order WENO schemes Shu (1998), could get less absolute numerical errors than the classical same order WENO schemes, and compress nonphysical oscillations nearby strong shocks or contact discontinuities. Some benchmark tests are performed to illustrate the capability of these schemes.

Enhanced Robustness of the Hybrid Compact-WENO Finite Difference Scheme for Hyperbolic Conservation Laws with Multi-resolution Analysis and Tukey’s Boxplot Method

The high order shock detection algorithm employing the high order multi-resolution (MR) analysis (Harten in J Comput Phys 49:357–393, 1983) for identifying the smooth and non-smooth stencils has been employed extensively in the hybrid schemes, such as the hybrid compact-weighted essentially non-oscillatory finite difference scheme, for solving hyperbolic conservation laws containing both discontinuous and complex fine scale structures. However, the accuracy and efficiency of the hybrid scheme depend heavily on the setting of a user defined ad hoc and problem dependent parameter, namely, MR tolerance $$\epsilon _{\scriptscriptstyle {MR}}$$ . In this study, we improve the outlier-detection algorithm based on the Tukey’s boxplot method as proposed by Vuik et al. (SIAM J Sci Comput 38:A84–A104, 2016) for solving the hyperbolic conservation laws with the hybrid scheme. By incorporating the global mean of the data in the definition of fences in each segmented subdomain, spurious false positive of the discontinuities can be eliminated. The improved outlier-detection algorithm essentially removes the need of specifying the MR Tolerance $$\epsilon _{\scriptscriptstyle {MR}}$$ , and thus greatly improves the robustness of the hybrid scheme. The accuracy, efficiency and robustness of the improved hybrid scheme with the shock detection algorithm using the high order MR analysis and the improved outlier-detection algorithm are demonstrated with numerous classical one- and two-dimensional examples of the shallow water equations and the Euler equations with discontinuous solutions.

Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations

In this paper, a class of high-order central Hermite WENO (HWENO) schemes based on finite volume framework and staggered meshes is proposed for directly solving one- and two-dimensional Hamilton-Jacobi (HJ) equations. The methods involve the Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. This work can be regarded as an extension of central HWENO schemes for hyperbolic conservation laws (Tao et al. J. Comput. Phys. 318, 222–251, 2016) which combine the central scheme and the HWENO spatial reconstructions and therefore carry many features of both schemes. Generally, it is not straightforward to design a finite volume scheme to directly solve HJ equations and a key ingredient for directly solving such equations is the reconstruction of numerical Hamiltonians to guarantee the stability of methods. Benefited from the central strategy, our methods require no numerical Hamiltonians. Meanwhile, the zeroth-order and the first-order moments of the solution are involved in the spatial HWENO reconstructions which is more compact compared with WENO schemes. The reconstructions are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. A collection of one- and two- dimensional numerical examples is performed to validate high resolution and robustness of the methods in approximating the solutions of HJ equations, which involve linear, nonlinear, smooth, non-smooth, convex or non-convex Hamiltonians.

Central Weighted ENO Schemes for Hyperbolic Conservation Laws on Fixed and Moving Unstructured Meshes

We present a novel arbitrary high order accurate central WENO spatial reconstruction procedure (CWENO) for the solution of nonlinear systems of hyperbolic conservation laws on fixed and moving unstructured simplex meshes in two and three space dimensions. Starting from the given cell averages of a function on a triangular or tetrahedral control volume and its neighbors, the nonlinear CWENO reconstruction yields a high order accurate and essentially non-oscillatory polynomial that is defined everywhere in the cell. Compared to other WENO schemes on unstructured meshes, the total stencil size is the minimum possible one, as in classical point-wise WENO schemes of Jiang and Shu. However, the linear weights can be chosen arbitrarily, which makes the practical implementation on general unstructured meshes particularly simple. We make use of the piecewise polynomials generated by the CWENO reconstruction operator inside the framework of fully discrete and high order accurate one-step ADER finite volume schemes on fixed Eulerian grids as well as on moving Arbitrary-Lagrangian-Eulerian (ALE) meshes. The computational efficiency of the high order finite volume schemes based on the new CWENO reconstruction is tested on several two- and three-dimensional systems of hyperbolic conservation laws and is found to be more efficient in terms of memory consumption and computational efficiency with respect to classical WENO reconstruction schemes on unstructured meshes. We also provide evidence that the new algorithm is suitable for implementation on massively parallel distributed memory supercomputers, showing two numerical examples run with more than one billion degrees of freedom in space. To our knowledge, at present these are the largest simulations ever run with unstructured WENO finite volume schemes.

A high-order monotonicity-preserving scheme for hyperbolic conservation laws

This work aims at designing a numerical model for discretizing non-linear hyperbolic systems of conservation laws, with high-order accuracy, while verifying a positivity property of the discretization. From a finite-volume discretization of the PDEs, we identify three positivity constraints for the resulting numerical scheme to be a convex combination of the stencil supporting the discretization. These constraints are then introduced into the MUSCL-like least-square interpolation procedure by using limiters conveniently sized. Lastly, a high-order stability-preserving time integration method (SSPRK) enables to generate the final algorithm. Extensive numerical tests on scalar problems help to assess the benefits and potentialities of the resulting algorithm. Then, upon using a local diagonalization, we extend the scope of the algorithm to non linear hyperbolic systems. To this aim, we propose two strategies for identifying local directions that promotes this diagonalization. Numerical tests on standard problems of gas dynamics, help to appreciate the advantages and drawbacks of each strategy. The resulting numerical scheme is a fourth-order least-square positive scheme (LSQP4) and it is designed and formulated for discretizing unsteady multi-dimensional hyperbolic conservation laws, over non-Cartesian meshes.

An Approximate Lax–Wendroff-Type Procedure for High Order Accurate Schemes for Hyperbolic Conservation Laws

A high order time stepping applied to spatial discretizations provided by the method of lines for hyperbolic conservations laws is presented. This procedure is related to the one proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185---2198, 2003) for numerically solving hyperbolic conservation laws. Both methods are based on the conversion of time derivatives to spatial derivatives through a Lax---Wendroff-type procedure, also known as Cauchy---Kovalevskaya process. The original approach in Qiu and Shu (2003) uses the exact expressions of the fluxes and their derivatives whereas the new procedure computes suitable finite difference approximations of them ensuring arbitrarily high order accuracy both in space and time as the original technique does, with a much simpler implementation and generically better performance, since only flux evaluations are required and no symbolic computations of flux derivatives are needed.

A Runge–Kutta discontinuous Galerkin scheme for hyperbolic conservation laws with discontinuous fluxes

The paper proposes a scheme by combining the Runge–Kutta discontinuous Galerkin method with a δ-mapping algorithm for solving hyperbolic conservation laws with discontinuous fluxes. This hybrid scheme is particularly applied to nonlinear elasticity in heterogeneous media and multi-class traffic flow with inhomogeneous road conditions. Numerical examples indicate the scheme’s efficiency in resolving complex waves of the two systems. Moreover, the discussion implies that the so-called δ-mapping algorithm can also be combined with any other classical methods for solving similar problems in general.

An Analysis on Induced Numerical Oscillations by Lax-Friedrichs Scheme

Induced numerical oscillations in the computed solution by monotone schemes for hyperbolic conservation laws has been a focus of recent studies. In this work using a local maximum principle, the monotone stable Lax-Friedrichs (LxF) scheme is investigated to explore the cause of induced local oscillations in the computed solution. It expounds upon that LxF scheme is locally unstable and therefore exhibits induced such oscillations. The carried out analysis gives a deeper insight to characterize the type of data extrema and the solution region which cause local oscillations. Numerical results for benchmark problems are also given to support the theoretical claims.

Numerical schemes for networks of hyperbolic conservation laws

In this paper we propose a procedure to extend classical numerical schemes for hyperbolic conservation laws to networks of hyperbolic conservation laws. At the junctions of the network we solve the given coupling conditions and minimize the contributions of the outgoing numerical waves. This flexible procedure allows us to also use central schemes at the junctions. Several numerical examples are considered to investigate the performance of this new approach compared to the common Godunov solver and exact solutions.

Seventh order Hermite WENO scheme for hyperbolic conservation laws

In this paper, we construct a new seventh-order Hermite weighted essentially non-oscillatory (HWENO7) scheme, for solving one and two dimensional hyperbolic conservation laws, by extending the fifth order HWENO introduced in Qiu J, Shu C-W. J Comput Phys 2003;193:115–135. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both function and its derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. The major advantage of HWENO schemes is its compactness in the reconstruction. For example, seven points are needed in the stencil for a seventh order WENO reconstruction, while only five points are needed for HWENO7 reconstruction. We use the central-upwind flux [Kurganov A, Noelle S, Petrova G, SIAM J Sci Comp 2001;23:707–740] which is simple, universal and efficient. The numerical solution is advanced in time by the seventh order linear strong-stability-preserving Runge–Kutta (ℓSSPRK) scheme for linear problems and the fourth order SSPRK(5,4) for nonlinear problems. The resulting scheme improves the convergence and accuracy of smooth parts of solution as well as decrease the dissipation near discontinuities. This is especially for long time evolution problems. Numerical experiments of the new scheme for one and two dimensional problems are reported. The results demonstrate that the proposed scheme is superior to the original HWENO and classical WENO schemes.

A modified fifth-order WENOZ method for hyperbolic conservation laws

The paper analyses by Taylor series the several fifth-order of accuracy schemes for hyperbolic conservation laws: the classical WENOJS scheme Jiang and Shu (1996), the WENOM scheme Henrick et al. (2005), the WENOZ scheme Borges et al. (2008) and the scheme, called WENOε here, Aràndiga et al. (2011). The order of weights of these four schemes agreed to the optimal weights is presented in detail. Then three prerequisites are developed if one intends to improve the WENOJS scheme: the scheme arrives the 5th-order at critical points; the weights of scheme approximate the optimal weights with high-order accuracy when solution is smooth; the scheme should not introduce much oscillations intuitively in the vicinity of discontinuities. According to the prerequisites above, a new WENO scheme (MWENOZ) is devised which is similar to the WENOZ scheme. Finally, the method designed here is demonstrated robustly by applying it to 1D and 2D numerical simulations and its advantage compared with the WENOZ scheme seems more striking in 2D problems.

A third-order entropy stable scheme for hyperbolic conservation laws

A third-order entropy stable scheme for nonlinear hyperbolic conservation laws is proposed here. This scheme contains two main ingredients: a fourth-order entropy conservative flux and a third-order numerical diffusion operator. A piecewise-quadratic reconstruction from pointwise values is developed in order to approximate the third-order dissipative term. To guarantee a non-oscillating property, a nonlinear limiter is employed and, furthermore, the scheme is proven to be entropy stable. Finally, numerical experiments are presented and demonstrate the accuracy, high-resolution, and robustness of our method.

Analysis of Hybrid Upwinding for Fully-Implicit Simulation of Three-Phase Flow with Gravity

The systems of algebraic equations arising from implicit (backward-Euler) finite-volume discretization of the conservation laws governing multiphase flow in porous media are quite challenging for nonlinear solvers. In the presence of countercurrent flow due to buoyancy, the numerical flux obtained with single-point Phase-Potential Upwinding (PPU) is not differentiable, which causes convergence difficulties for nonlinear solvers. Recently, [Lee, Efendiev, and Tchelepi, Adv. Water Resour., 82 (2015), pp. 27--38] proposed a hybrid upwinding strategy for two-phase flow that gives rise to a differentiable numerical flux across the entire viscous-gravity parameter space. Here, we first present an Implicit Hybrid Upwinding (IHU) scheme for hyperbolic conservation laws, extending the work of Lee, Efendiev, and Tchelepi to an arbitrary number of fluid phases. We show that the numerical flux obtained with the IHU is consistent, and a monotone function of its own saturation. It is also a differentiable function of t...

A discontinuous wave-in-cell numerical scheme for hyperbolic conservation laws

A new Riemann solver scheme for hyperbolic systems is introduced. The method consists of a discretization of the initial data into an approximate representation by discrete, discontinuous waves. Instead of calculating an intercell flux based on these waves, the discontinuous waves are propagated directly. Since the sum total of all discontinuous waves represents an extension of the linear Riemann problem, the solution is determined straightforwardly. For nonlinear systems, each timestep is considered a separate linear Riemann problem, and the projected waves are weighted to a background grid. This method is strikingly similar to the particle-in-cell approach, except that discontinuous waves are pushed around instead of macroparticles. The method is applied to Maxwell's equations and the equations of inviscid gasdynamics. For linear systems, exact solutions can be achieved without dissipation, and exact transmissive boundary condition treatments are trivial to implement. For inviscid gasdynamics, the nonlinear method tended to resolve discontinuities more sharply than the Roe, HLL or HLLC methods while requiring only between 30% and 50% of the execution time under identical conditions. The approach is also extremely robust, as it works for any Courant number. In the limit that the Courant number becomes infinite, the nonlinear solution approaches the linearized solution.

High-order accurate, fully discrete entropy stable schemes for scalar conservation laws

The recently developed TECNO schemes for hyperbolic conservation laws are designed to be high-order accurate and entropy stable, but are, as of yet, only semidiscrete. We perform an explicit discretization of the temporal derivative to obtain a fully discrete scheme, and derive a rather unrestrictive CFL condition that ensures global entropy stability. The scheme is tested in a series of numerical experiments.

Stability and convergence of relaxation schemes to hyperbolic balance laws via a wave operator

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505–527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357–1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277] for systems of hyperbolic conservation laws without source terms.

A high‐resolution hybrid scheme for hyperbolic conservation laws

SummaryA new hybrid scheme is proposed, which combines the improved third‐order weighted essentially non‐oscillatory (WENO) scheme presented in this paper with a fourth‐order central scheme by a novel switch. Two major steps have been gone through for the construction of a high‐performance and stable hybrid scheme. Firstly, to enhance the WENO part of the hybrid scheme, a new reference smoothness indicator has been devised, which, combined with the nonlinear weighting procedure of WENO‐Z, can drive the third‐order WENO toward the optimal linear scheme faster. Secondly, to improve the hybridization with the central scheme, a hyperbolic tangent hybridization switch and its efficient polynomial counterpart are devised, with which we are able to fix the threshold value introduced by the hybridization. The new hybrid scheme is thus formulated, and a set of benchmark problems have been tested to verify the performance enhancement. Numerical results demonstrate that the new hybrid scheme achieves excellent performance in resolving complex flow features, even compared with the fifth‐order classical WENO scheme and WENO‐Z scheme. Copyright © 2015 John Wiley & Sons, Ltd.

Entropy production and mesh refinement – application to wave breaking

We propose an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. This approach is validated in the case of classical 1D numerical tests. Then, we present a 3D application. Indeed, according to an eulerian bi-fluid formulation at low Mach, an hyperbolic system of conservation laws allows an easily parallelization and mesh refinement by blocks .

High-order central ENO finite-volume scheme for hyperbolic conservation laws on three-dimensional cubed-sphere grids

A fourth-order accurate finite-volume scheme for hyperbolic conservation laws on three-dimensional (3D) cubed-sphere grids is described. The approach is based on a central essentially non-oscillatory (CENO) finite-volume method that was recently introduced for two-dimensional compressible flows and is extended to 3D geometries with structured hexahedral grids. Cubed-sphere grids feature hexahedral cells with nonplanar cell surfaces, which are handled with high-order accuracy using trilinear geometry representations in the proposed approach. Varying stencil sizes and slope discontinuities in grid lines occur at the boundaries and corners of the six sectors of the cubed-sphere grid where the grid topology is unstructured, and these difficulties are handled naturally with high-order accuracy by the multidimensional least-squares based 3D CENO reconstruction with overdetermined stencils. A rotation-based mechanism is introduced to automatically select appropriate smaller stencils at degenerate block boundaries, where fewer ghost cells are available and the grid topology changes, requiring stencils to be modified. Combining these building blocks results in a finite-volume discretization for conservation laws on 3D cubed-sphere grids that is uniformly high-order accurate in all three grid directions. While solution-adaptivity is natural in the multi-block setting of our code, high-order accurate adaptive refinement on cubed-sphere grids is not pursued in this paper. The 3D CENO scheme is an accurate and robust solution method for hyperbolic conservation laws on general hexahedral grids that is attractive because it is inherently multidimensional by employing a K-exact overdetermined reconstruction scheme, and it avoids the complexity of considering multiple non-central stencil configurations that characterizes traditional ENO schemes. Extensive numerical tests demonstrate fourth-order convergence for stationary and time-dependent Euler and magnetohydrodynamic flows on cubed-sphere grids, and robustness against spurious oscillations at 3D shocks. Performance tests illustrate efficiency gains that can be potentially achieved using fourth-order schemes as compared to second-order methods for the same error level. Applications on extended cubed-sphere grids incorporating a seventh root block that discretizes the interior of the inner sphere demonstrate the versatility of the spatial discretization method.