This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A new strategy is introduced here that allows non-conservative products to be written as the derivative of a generalized flux function that is defined locally on the basis of the selected family of paths. WENO reconstructions are then applied to this generalized flux. Moreover, if a Roe linearization is available, the generalized flux function can be evaluated through matrix vector operations instead of path-integrals. Two different known techniques are used to extend the methods to problems with source terms and the well-balanced properties of the resulting schemes are studied. These numerical schemes are applied to a coupled Burgers system and to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.
The decisive factor for the calculation accuracy of the mapped weighted essentially non-oscillatory scheme is the width of the center region of the mapping function. Through analysis of the classical mapped WENO schemes, the results show the width of the central range of the mapping function determined by the local operator in its denominator. Substituting the local operator in WENO-AIM with a symmetric one and an asymmetric function, we get two new adaptive mapped WENO schemes, WENO-AIMS and WENO-AIMA. Similarly, we improve WENO-RM260 and WENO-PM6 by using these local operators, and composed …
Hyperbolic systems under nonconservative form arise in numerous applications modeling physical processes, for example from the relaxation of more general equations (e.g. with dissipative terms). This paper reviews an existing class of augmented Roe schemes and discusses their application to linear nonconservative hyperbolic systems with source terms. We extend existing augmented methods by redefining them within a common framework which uses a geometric reinterpretation of source terms. This results in intrinsically well-balanced numerical discretizations. We discuss two equivalent formulation…
The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important ques…
In this paper we propose new Z-type nonlinear weights of the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme for hyperbolic conservation laws. Instead of employing the classical smoothness indicators for the nonlinear weights, we take the $p$th root of the smoothness indicators and follow the form of Z-type nonlinear weights, leading to fifth order accuracy in smooth regions, even at the critical points, and sharper approximations around the discontinuities. We also prove that the proposed nonlinear weights converge to the linear weights as $p \to \infty$, impl…
The advantage of WENO-JS5 scheme [ J. Comput. Phys. 1996] over the WENO-LOC scheme [J. Comput. Phys.1994] is that the WENO-LOC nonlinear weights do not achieve the desired order of convergence in smooth monotone regions and at critical points. In this article, this drawback is achieved with the WENO-LOC smoothness indicators by constructing a WENO-Z type nonlinear weights which contains a novel global smoothness indicator. This novel smoothness indicator measures the derivatives of the reconstructed flux in a global stencil, as a result, the proposed numerical scheme could decrease the dissipa…
Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. Furthermore, good properties of fixed-point iterative sweeping methods include that they have explicit forms and do not involve inverse operation of nonlinear local systems, and they can be applied to general hyperbolic equations using a…
A fully convergent fixed-point fast sweeping method with the WENO-JS local solver for steady state of hyperbolic conservation laws
School of Mathematics and Statistics, Huang Huai University（河南工程学院数学与统计学院）；College of Science and Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing University of Aeronautics and Astronautics（南京航空航天大学理学院及工信部航空器数学建模与高性能计算重点实验室）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame（圣母大学应用与计算数学和统计系）
A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所计算物理实验室）
In this paper, we propose a hybrid finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for solving one and two dimensional hyperbolic conservation laws. The zeroth-order and the first-order moments are used in the spatial reconstruction, with total variation diminishing Runge-Kutta time discretization. The main idea of the hybrid HWENO scheme is that we first use a shock-detection technique to identify the troubled cell, then, if the cell is identified as a troubled cell, we would modify the first order moment in the troubled cell and employ HWENO reconstruction in spatia…
In this paper we extensively study the stochastic Galerkin scheme for uncertain systems of conservation laws, which appears to produce oscillations already for a simple example of the linear advection equation with Riemann initial data. Therefore, we introduce a modified scheme that we call the weighted essentially non-oscillatory (WENO) stochastic Galerkin scheme, which is constructed to prevent the propagation of Gibbs phenomenon into the stochastic domain by applying a slope limiter in the stochasticity. In order to achieve a high order method, we use a spatial WENO reconstruction and also …
A new type of finite volume WENO schemes for hyperbolic problems was devised in [36] by introducing the order-preserving (OP) criterion. In this continuing work, we extend the OP criterion to the WENO-Z-type schemes. We firstly rewrite the formulas of the Z-type weights in a uniform form from a mapping perspective inspired by extensive numerical observations. Accrodingly, we build the concept of the locally order-preserving (LOP) mapping which is an extension of the order-preserving (OP) mapping and the resultant improved WENO-Z-type schemes are denoted as LOP-GMWENO-X. There are four major ad…
In this paper, a fifth-order Hermite weighted essentially non-oscillatory (HWENO) scheme with artificial linear weights is proposed for one and two dimensional hyperbolic conservation laws, where the zeroth-order and the first-order moments are used in the spatial reconstruction. We construct the HWENO methodology using a nonlinear convex combination of a high degree polynomial with several low degree polynomials, and the associated linear weights can be any artificial positive numbers with only requirement that their summation equals one. The one advantage of the HWENO scheme is its simplicit…
We propose an alternative reconstruction for weighted essentially non-oscillatory schemes with adaptive order (WENO-AO) for solving hyperbolic conservation laws. The alternative reconstruction has a more concise form than the original WENO-AO reconstruction. Moreover, it is a strictly convex combination of polynomials with unequal degrees. Numerical examples show that the alternative reconstruction maintains the accuracy and robustness of the WENO-AO schemes.
Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems
Laboratoire de Mathématiques Jean Leray, Centre National de la Recherche Scientifique, Université de Nantes（让·勒雷数学实验室，法国国家科学研究中心，南特大学）；Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室，法国国家科学研究中心，皮埃尔和玛丽·居里大学（巴黎第六大学））
