Second-order schemes for conservation laws with discontinuous flux modelling clarifier–thickener units

Continuously operated clarifier–thickener (CT) units can be modeled by a non-linear, scalar conservation law with a flux that involves two parameters that depend discontinuously on the space variable. This paper presents two numerical schemes for the solution of this equation that have formal second-order accuracy in both the time and space variable. One of the schemes is based on standard total variation diminishing (TVD) methods, and is addressed as a simple TVD (STVD) scheme, while the other scheme, the so-called flux-TVD (FTVD) scheme, is based on the property that due to the presence of the discontinuous parameters, the flux of the solution (rather than the solution itself) has the TVD property. The FTVD property is enforced by a new nonlocal limiter algorithm. We prove that the FTVD scheme converges to a BV _t solution of the conservation law with discontinuous flux. Numerical examples for both resulting schemes are presented. They produce comparable numerical errors, while the FTVD scheme is supported by convergence analysis. The accuracy of both schemes is superior to that of the monotone first-order scheme based on the adaptation of the Engquist–Osher scheme to the discontinuous flux setting of the CT model (Bürger, Karlsen and Towers in SIAM J Appl Math 65:882–940, 2005). In the CT application there is interest in modelling sediment compressibility by an additional strongly degenerate diffusion term. Second-order schemes for this extended equation are obtained by combining either the STVD or the FTVD scheme with a Crank–Nicolson discretization of the degenerate diffusion term in a Strang-type operator splitting procedure. Numerical examples illustrate the resulting schemes.     

On the convergence of a second order approximation of conservation laws with discontinuous flux

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to the (A,B) entropysolution. Numerical results are presented to validate the convergence analysis. In addition, it is inferred that these numerical solutions are comparable with those obtained using the standard minmod limiter.     

ON ENTROPY CONDITIONS OF HIGH RESOLUTION SCHEMES FOR SCALAR CONSERVATION LAWS

1. IntroductionLet us consider the Cauchy problems for nonlinear hroerbolic scalar cO~ion laws:where the function f(u) E C'(R) and the hatal data ho E BV(R). As is well lmown, thisProblem in general does not ~ smooth solution, so that weak solution in the sense ofdistributions must be considered. Moreover, an elitropy condition must be added in order toensure the ~s of the weak solutions.The research of numerical methods for solving the equation (1.1) has been developed rapidlyin this deca…     

An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations

In this article, we introduce an invariant‐region‐preserving (IRP) limiter for the p‐system and the corresponding viscous p‐system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For high order discontinuous Galerkin (DG) schemes to the p‐system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscous p‐system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes.     

Symmetrical weighted essentially non‐oscillatory‐flux limiter schemes for Hamilton–Jacobi equations

In this paper, we propose a new scheme that combines weighted essentially non‐oscillatory (WENO) procedures together with monotone upwind schemes to approximate the viscosity solution of the Hamilton–Jacobi equations. In one‐dimensional (1D) case, first, we obtain an optimum polynomial on a four‐point stencil. This optimum polynomial is third‐order accurate in regions of smoothness. Next, we modify a second‐order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction‐evolution method limiter. Finally, the optimum polynomial is considered as a symmetric and convex combination of three polynomials with ideal weights. Following the methodology of the classic WENO procedure, then, we calculate the non‐oscillatory weights with the ideal weights. Numerical experiments in 1D and 2D are performed to compare the capability of the hybrid scheme to WENO schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

An Engquist-Osher type finite difference scheme with a discontinuous flux function in space

An Engquist-Osher type finite difference scheme is derived for dealing with scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. The new monotone difference scheme is based on introducing a new interface numerical flux function, which is called a generalized Engquist-Osher flux. By means of this scheme, the existence and uniqueness of weak solutions to the scalar conservation laws are obtained and the convergence theorem is established. Some numerical examples are presented and the corresponding numerical results are displayed to illustrate the efficiency of the methods.     

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     

求解双曲型方程的一致二阶守恒型差分格式的构造及数值实验

由于TVD格式具有激波分辨率高与非物理振荡小的特点,在气体动力学问题的求解中得到了广泛的应用.但现有的TVD格式受其构造方式所限,在解的局部极值点附近只能达到一阶精度. 考虑以下单个双曲型方程:     

非结构网格上一类满足局部极值原理的三阶精度有限体积方法

对二维标量双曲型守恒律方程,发展了一类满足局部极值原理的非结构网格有限体积格式.其构造思想是,以单调数值通量为基础,通过应用基于最小二乘法的二次重构和极值限制器,使数值解满足局部极值原理.为保证数值解在光滑区域达到三阶精度,该格式可结合局部光滑探测器使用.本文从理论上分析了格式的稳定性条件,数值实验验证了格式的精度和对间断的分辨能力.     

Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.     

High Resolution Schemes and Discrete Entropy Conditions for 2-D Linear Conservation Laws

In this paper, fully discrete entropy conditions of a class of high resolution schemes with the MmB property are discussed by using the theory of proper discrete entropy flux for the linear scalar conservation laws in two dimensions. The theoretical resluts show that the high resolution schemes satisfying fully discrete entropy conditions with proper discrete entropy flux cannot preserve second order accuracy in the case of two dimensions.     

Convergence of higher order finite volume schemes on irregular grids

We prove convergence to the entropy solution of a general class of higher order finite volume schemes on unstructured, irregular grids for multidimensional scalar conservation laws. Such grids allow for cells to become flat in the limit. We derive a new entropy inequality for higher order schemes built on Godunov’s numerical flux. Our result implies convergence of suitably modified versions of MUSCL-type finite volume schemes, ENO schemes and the discontinuous Galerkin finite element method.     

A Large Time Step explicit scheme (CFL>1) on unstructured grids for 2D conservation laws: Application to the homogeneous shallow water equations

In this work, a Large Time Step (LTS) explicit finite volume scheme designed to allow CFL > 1 is applied to the numerical resolution of 2D scalar and systems of conservation laws on triangular grids. Based on the flux difference splitting formulation, a special concern is put on finding the way of packing the information to compute the numerical solution when working on unstructured grids. Not only the cell areas but also the length of the interfaces and their orientation are questions of interest to send the information from each edge or interface. The information to update the cell variables is computed according to the local average discrete velocity and the orientation of the edges of the cells involved. The performance of these ideas is tested and compared with the conventional explicit first order and second order schemes in academic configurations for the 2D linear scalar equation and for 2D systems of conservation laws (in particular the shallow water equations) without source terms. The LTS scheme is demonstrated to preserve or even gain accuracy and save computational time with respect to the first order scheme.     

Comparison of two cell-centered multigrid schemes for problems with discontinuous coefficients

Two different schemes for constructing coarse-grid operators are implemented in a linear multigrid code. In the first scheme, the construction of the coarse-grid operators is done using a variational approach. Certain conservation properties of the fine-grid matrices are shown to be preserved on the coarser grids by the variational construction. In the second scheme, the diffusion coefficients for the coarse grids are calculated by a simple restriction of the coefficient from the fine grid, using a flux conservation principle. The multigrid codes are then applied to solve the linear equations from an IMPES formulation of a two-phase porous-media flow model. A standard elliptic model problem with jump discontinuous coefficients is also solved using the two multigrid schemes. In simple cases of particular elliptic equations these two schemes are identical. However, in more general cases, such as in reservoir problems, these schemes differ. It is shown that multigrid efficiency typical of the constant coefficient cases is obtained for these problems involving discontinuous coefficients. © 1993 John Wiley & Sons, Inc.     

Convergence of higher order finite volume schemes on irregular grids

We prove convergence to the entropy solution of a general class of higher order finite volume schemes on unstructured, irregular grids for multidimensional scalar conservation laws. Such grids allow for cells to become flat in the limit. We derive a new entropy inequality for higher order schemes built on Godunov's numerical flux. Our result implies convergence of suitably modified versions of MUSCL-type finite volume schemes, ENO schemes and the discontinuous Galerkin finite element method. Supported by Deutsche Forschungsgemeinschaft, SFB256.     

MUSTA: A multi-stage numerical flux

Numerical methods for conservation laws constructed in the framework of finite volume and discontinuous Galerkin finite elements require, as the building block, a monotone numerical flux. In this paper we present some preliminary results on the MUSTA approach [E.F. Toro, Multi-stage predictor–corrector fluxes for hyperbolic equations, Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June, 2003] for constructing upwind numerical fluxes. The scheme may be interpreted as an un-conventional approximate Riemann solver that has simplicity and generality as its main features. When used in its first-order mode we observe that the scheme achieves the accuracy of the Godunov method used in conjunction with the exact Riemann solver, which is the reference first-order method for hyperbolic systems. At least for the scalar model hyperbolic equation, the Godunov scheme is the best of all first-order monote schemes, it has the smallest truncation error. Extensions of the scheme of this paper are realized in the framework of existing approaches. Here we present a second-order TVD (TVD for the scalar case) extension and show numerical results for the two-dimensional Euler equations on non-Cartesian geometries. The schemes find their best justification when solving very complex systems for which the solution of the Riemann problem, in the classical sense, is too complex, too costly or is simply unavailable.     

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework on unstructured meshes

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [39], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight or curved edges. This is an extension of our earlier work [4] in which the WENO limiter was designed for the CPR framework on regular meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper uses information only from the target cell and its immediate neighbors. Hence, it is particularly simple to implement and will not harm the conservativeness and compactness of the CPR framework. Since the CPR framework with this WENO limiter does not in general satisfy the positivity preserving property, we also extend the positivity-preserving limiters [36], [33] to the CPR framework. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of this procedure.