Convergence of relaxation schemes for conservation laws

We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L^∞ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.     

Centred TVD schemes for hyperbolic conservation laws

New first- and high-order centred methods for conservation lawsare presented. Convenient TVD conditions for constructing centredTVD schemes are then formulated and some useful results areproved. Two families of centred TVD schemes are constructedand extended to nonlinear systems. Some numerical results arealso presented.     

An efficient WENO scheme for solving hyperbolic conservation laws

In this paper we propose a new WENO scheme, in which we use a central WENO [G. Capdeville, J. Comput. Phys. 227 (2008) 2977-3014] (CWENO) reconstruction combined with the smoothness indicators introduced in [R. Borges, M. Carmona, B. Costa, W. Sun Don, J. Comput. Phys. 227 (2008) 3191-3211] (IWENO). We use the central-upwind flux [A. Kurganov, S. Noelle, G. Petrova, SIAM J. Sci. Comp. 23 (2001) 707-740] which is simple, universal and efficient. For time integration we use the third order TVD Runge-Kutta scheme. The resulting scheme improves the convergence order at critical points of smooth parts of solution as well as decrease the dissipation near discontinuities. Numerical experiments of the new scheme for one and two-dimensional problems are reported. The results demonstrates that the proposed scheme is superior to the original CWENO and IWENO schemes.     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

On the optimization of flux limiter schemes for hyperbolic conservation laws

A classic strategy to obtain high‐quality discretizations of hyperbolic partial differential equations is to use flux limiter (FL) functions for blending two types of approximations: a monotone first‐order scheme that deals with discontinuous solution features and a higher order method for approximating smooth solution parts. In this article, we study a new approach to FL methods. Relying on a classification of input data with respect to smoothness, we associate specific basis functions with the individual smoothness notions. Then, we construct a limiter as a linear combination of the members of parameter‐dependent families of basis functions, and we explore the possibility to optimize the parameters in interesting model situations to find a corresponding optimal limiter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Implicit kinetic schemes for scalar conservation laws

Based on kinetic formulation for scalar conservation laws, we present implicit kinetic schemes. For time stepping these schemes require resolution of linear systems of algebraic equations. The scheme is conservative at steady states. We prove that if time marching procedure converges to some steady state solution, then the implicit kinetic scheme converges to some entropy steady state solution. We give sufficient condition of the convergence of time marching procedure. For scalar conservation laws with a stiff source term we construct a stiff numerical scheme with discontinuous artificial viscosity coefficients that ensure the scheme to be equilibrium conserving. We couple the developed implicit approach with the stiff space discretization, thus providing improved stability and equilibrium conservation property in the resulting scheme. Numerical results demonstrate high computational capabilities (stability for large CFL numbers, fast convergence, accuracy) of the developed implicit approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 26–43, 2002     

An efficient TVD‐WENO method for conservation laws

In this article, we present a high‐resolution hybrid scheme for solving hyperbolic conservation laws in one and two dimensions. In this scheme, we use a cheap fourth order total variation diminishing (TVD) scheme for smooth region and expensive seventh order weighted nonoscillatory (WENO) scheme near discontinuities. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multiresolution technique. For time integration, we use the third order TVD Runge‐Kutta scheme. The accuracy of the resulting hybrid high order scheme is comparable with these of WENO, but with significant decrease of the CPU cost. Numerical demonstrates that the proposed scheme is comparable to the high order WENO scheme and superior to the fourth order TVD scheme. Our scheme has the added advantage of simplicity and computational efficiency. Numerical tests are presented which show the robustness and effectiveness of the proposed scheme.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Finite-difference schemes for solving multidimensional hyperbolic equations and their systems

A numerical algorithm for integrating second-order multidimensional hyperbolic equations and hyperbolic systems is described. Conditionally and unconditionally stable finite-difference schemes are constructed. The analysis of the schemes is based on the general regularization principle proposed by A.A. Samarskii.     

Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.     

BV-estimates of Lax–Friedrichs' scheme for hyperbolic conservation laws with relaxation

In this paper, we will give BV-estimates of Lax–Friedrichs' scheme for a simple hyperbolic system of conservation laws with relaxation and get the global existence and uniqueness of BV-solution by the BV-estimates above. Furthermore, our results show that the solution converge towards the solution of an equilibrium model as the relaxation time ε>0 tends to zero provided sub-characteristic condition holds. Copyright © 2007 John Wiley & Sons, Ltd.     

An adaptive wavelet viscosity method for hyperbolic conservation laws

We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

An adaptive mesh method for 1D hyperbolic conservation laws

An adaptive method is developed for solving one-dimensional systems of hyperbolic conservation laws, which combines the rezoning approach with the finite volume weighted essentially non-oscillatory (WENO) scheme. An a posteriori error estimate, used to equidistribute the mesh, is obtained from the differences between respective numerical solutions of 5th-order WENO (WENO5) and 3rd-order ENO (ENO3) schemes. The number of grids can be adaptively readjusted based on the solution structure. For higher efficiency, mesh readjustment is performed every few time steps rather than every time step. In addition, a high order conservative interpolation is used to compute the physical solutions on the new mesh from old mesh based on the finite volume ENO reconstruction. Extensive examples suggest that this adaptive method exhibits more accurate resolution of discontinuities for a similar level of computational time comparing with that on a uniform mesh.     

用WENO方法求解双曲型守恒律方程组的初(边)值问题

本文用WENO算法解决双曲型守恒律方程组初（边值）问题．给出一种满足熵条件、Sδ熵条件和边界熵条件的WENO算法．通过这个算法就能得到守恒律方程组的数值解，数值解和理论解是非常吻合的．     

Nonlinear diffusive-dispersive limits for scalar multidimensional conservation laws

We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we show that the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation law. Our proof is based on methodology developed in [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254] which uses the averaging lemma.     

A composite semi-conservative scheme for hyperbolic conservation laws

In this work a first order accurate semi-conservative composite scheme is presented for hyperbolic conservation laws. The idea is to consider the non-conservative form of conservation law and utilize the explicit wave propagation direction to construct semi-conservative upwind scheme. This method captures the shock waves exactly with less numerical dissipation but generates unphysical rarefaction shocks in case of expansion waves with sonic points. It shows less dissipative nature of constructed scheme. In order to overcome it, we use the strategy of composite schemes. A very simple criteria based on wave speed direction is given to decide the iterations. The proposed method is applied to a variety of test problems and numerical results show accurate shock capturing and higher resolution for rarefaction fan.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Boundary layer to a system of viscous hyperbolic conservation laws

In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for n × n hyperbolic system of conservation laws with artificial viscosity in the half line （0, ∞）. We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.     

Weakly nonoscillatory schemes for scalar conservation laws

A new class of Godunov-type numerical methods (called here weakly nonoscillatory or WNO) for solving nonlinear scalar conservation laws in one space dimension is introduced. This new class generalizes the classical nonoscillatory schemes. In particular, it contains modified versions of Min-Mod and UNO. Under certain conditions, convergence and error estimates for WNO methods are proved.

     

On the linearization of systems of conservation laws for fluids at a material contact discontinuity

In order to investigate the linearized stability or instability of compressible flows, as it occurs for instance in Rayleigh–Taylor or Kelvin–Helmholtz instabilities, we consider the linearization at a material discontinuity of a flow modeled by a multidimensional nonlinear hyperbolic system of conservation laws. Restricting ourselves to the plane-symmetric case, the basic solution is thus a one-dimensional contact discontinuity and the normal modes of pertubations are solutions of the resulting linearized hyperbolic system with discontinuous nonconstant coefficients and source terms. While in Eulerian coordinates, the linearized Cauchy problem has no solution in the class of functions, we prove that for a large class of systems of conservation laws written in Lagrangian coordinates and including the Euler and the ideal M.H.D. systems, there exists a unique function solution of the problem that we construct by the method of characteristics.