Localization effects and measure source terms in numerical schemes for balance laws

This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

     

Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.     

A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models

The paper is concerned with development of a new finite-volume method for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. The first step in the derivation of the new method is made by adding an equation for the chemoattractant concentration gradient to the original system. We then show that the convective part of the resulting system is typically of a mixed hyperbolic-elliptic type and therefore straightforward numerical methods for the studied system may be unstable. The proposed method is based on the application of the second-order central-upwind scheme, originally developed for hyperbolic systems of conservation laws in Kurganov et al. (SIAM J Sci Comput 21:707–740, 2001), to the extended system of PDEs. We show that the proposed second-order scheme is positivity preserving, which is a very important stability property of the method. The scheme is applied to a number of two-dimensional problems including the most commonly used Keller–Segel chemotaxis model and its modern extensions as well as to a haptotaxis system modeling tumor invasion into surrounding healthy tissue. Our numerical results demonstrate high accuracy, stability, and robustness of the proposed scheme.     

Kinetic approximation of a boundary value problem for conservation laws

Summary. We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.Mathematics Subject Classification (2000): 65M06, 65M12, 76M20Correspondence to: D. Aregba-Driollet     

An efficient splitting technique for two‐layer shallow‐water model

We consider the numerical approximation of the weak solutions of the two‐layer shallow‐water equations. The model under consideration is made of two usual one‐layer shallow‐water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one‐layer shallow‐water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow‐water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non‐negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1396–1423, 2015     

GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM北大核心CSCD

Based on the corrected finite pointset method (CFPM) with CPU-GPU heteroid parallelization (CFPM-GPU), a high-efficiency, accurate and fast parallel algorithm was developed for the high-dimensional phase separation phenomena governed by the multi-component Cahn-Hilliard (C-H) equation in complex domains. The proposed parallel algorithm with the CFPM-GPU was built in a process like: ① introduce the Wendland weight function into the discretization of the finite pointset method (FPM) scheme for the 1st/2nd spatial derivatives, based on the Taylor series and the weighted least square concept; ② use the above FPM scheme twice to approximate the 4th spatial derivative in the C-H equation, which is called the CFPM method; ③ for the first time establish an accelerating parallel algorithm for the CFPM with local matrices by means of a single GPU card based on the CUDA programming. Two benchmark problems of 2D radially and 3D spherically symmetric C-H equations were first solved to test the accuracy and high-efficiency of the proposed CFPM-GPU, and the acceleration ratio of the GPU parallelization to the single CPU computation is about 160. Subsequently, the proposed CFPM-GPU was used to predict the 2D/3D multi-phase separation phenomena in complex domains, and the prediction was compared with other numerical results. The numerical results show that, the proposed CFPM-GPU is valid and high-efficiency to simulate the 2D/3D multi-phase separation cases in complex domains. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

ON THE CENTRAL RELAXING SCHEMES I:SINGLE CONSERVATION LAWS

1. IntroductionIn [6], Jin and adn constructed a class of uPWind relaxing schemes for nonlinearconservation lawswith initial data u(0, x) ~ "o(x), x ~ (xl, ...t -cd), by using the idea of the local relaxation approximation [2,3,6,10].The relaxing scheme is obtained in the following way: A linear hyperbolic systemwith a stiff source term is first constructed to approximate the original equation (1.1)with a small dissipative correction. Then this linear hyperbolic system is solved easilyby und…     

Nonlinear generalized functions and jump conditions for a standard one pressure liquid–gas model

The mixture of a liquid and a gas is classically represented by one pressure models. These models are a system of PDEs in nonconservative form and shock wave solutions do not make sense within the theory of distributions: they give rise to products of distributions that are not defined within distribution theory. But they make sense by applying a theory of nonlinear generalized functions to these equations. In contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori. Jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations. The physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible, and so they do not satisfy equally well their respective state laws on the shock waves. This modelization gives well defined explicit jump conditions. The great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results. The jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions, which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results. We expose with care the mathematical originality of the theory of nonlinear generalized functions (an original abstract analysis issued by the Leopoldo Nachbin team on infinite dimensional holomorphy) that permits to state mathematically physical facts that cannot be formulated within distribution theory, and are the key for the removal of “ambiguities” that classically appear when one tries to calculate on “multiplications of distributions” that occur in the differential equations of physics.     

A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates

In this paper, we extend our previous work (M.-C. Lai, A simple compact fourth-order Poisson solver on polar geometry, J. Comput. Phys. 182 (2002) 337–345) to 3D cases. More precisely, we present a spectral/finite difference scheme for Poisson equation in cylindrical coordinates. The scheme relies on the truncated Fourier series expansion, where the partial differential equations of Fourier coefficients are solved by a formally fourth-order accurate compact difference discretization. Here the formal fourth-order accuracy means that the scheme is exactly fourth-order accurate while the poles are excluded and is third-order accurate otherwise. Despite the degradation of one order of accuracy due to the presence of poles, the scheme handles the poles naturally; thus, no pole condition is needed. The resulting linear system is then solved by the Bi-CGSTAB method with the preconditioner arising from the second-order discretization which shows the scalability with the problem size.     

An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.     

A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes

Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017     

Convergence des méthodes particulaires renormalisées pour les systèmes de Friedrichs

We present a study of the renormalized particle scheme. Renormalization is a tool introduced in order to alleviate the SPH particle methods' lack of consistency. A conservative scheme, the weak renormalized scheme, is derived from the general conservation laws weak formulation. We apply this scheme to Friedrichs systems. The weak renormalized scheme being unstable, we introduce a numerical viscosity before applying an explicit Euler time discretization, and thus construct the numerical scheme whose convergence in $L^2$ norm is studied. To cite this article: N. Lanson, J.-P. Vila, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

二相流计算的一种差分算法

考虑两相流的力学行为,忽略相间的耗散作用,建立了Euler型的基本控制方程．状态方程采用刚性状态方程．基于Abgrall提出的准则,在流动区域内,对可压两相流提出了一个精度较高的Euler型数值方法,数值格式是Godunov型格式,对守恒型和非守恒型方程采用HLLC型和Lax-Friedrichs型近似Riemann解算器,引入了速度驰豫和压强驰豫过程来代替两相间的相互作用．在一维情形下给出数值算例,并且和Saurel的算例进行了比较,结果表明该算法不但精确而且稳定,且在间断处没有数值振荡．     

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.     

Asymptotic preserving HLL schemes

This work concerns the derivation of HLL schemes to approximate the solutions of systems of conservation laws supplemented by source terms. Such a system contains many models such as the Euler equations with high friction or the M1 model for radiative transfer. The main difficulty arising from these models comes from a particular asymptotic behavior. Indeed, in the limit of some suitable parameter, the system tends to a diffusion equation. This article is devoted to derive HLL methods able to approximate the associated transport regime but also to restore the suitable asymptotic diffusive regime. To access such an issue, a free parameter is introduced into the source term. This free parameter will be a useful correction to satisfy the expected diffusion equation at the discrete level. The derivation of the HLL scheme for hyperbolic systems with source terms comes from a modification of the HLL scheme for the associated homogeneous hyperbolic system. The resulting numerical procedure is robust as the source term discretization preserves the physical admissible states. The scheme is applied to several models of physical interest. The numerical asymptotic behavior is analyzed and an asymptotic preserving property is systematically exhibited. The scheme is illustrated with numerical experiments. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1396–1422, 2011     

A combined mixed finite element ADI scheme for solving Richards' equation with mixed derivatives on irregular grids

We propose and analyze an efficient numerical method for solving semilinear parabolic problems with mixed derivative terms on non-rectangular domains. The spatial semidiscretization process is based on an expanded mixed finite element scheme which, combined with suitable quadrature rules, is converted into a cell-centered finite difference scheme. This choice preserves the asymptotic accuracy and local conservation of mass of the method, while substantially reducing the computational cost of the totally discrete scheme. To obtain it, an alternating direction implicit scheme is used for the integration in time. The resulting numerical algorithm involves sets of uncoupled tridiagonal systems which can be solved in parallel. We set out some theoretical results of unconditional convergence (of second order in space and first order in time) for our method. Finally, a numerical experiment is shown in order to illustrate the theoretical results.     

Computation of dynamical phase transitions in solids

This paper deals with the dynamics of phase boundaries in a nonlinear elastic two-phase material. We consider the elasticity system in 1D and the equations of anti-plane shear motion in 2D, where effects of viscosity and capillarity are neglected. These first-order conservation laws allow to represent phase boundaries as shock-like sharp interfaces. However, in contrast to what is known for homogeneous materials, the entropy inequality does not select a unique solution, and an additional criterion, the so-called kinetic relation, is required.Based on a scheme introduced by Hou, Rosakis and LeFloch [T. Hou, Ph. Rosakis, P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics, J. Comput. Phys. 150 (1999) 302–331] we focus on the numerical solution of a specific model system. Using a level-set technique to enforce the kinetic relation on the discrete level leads to a reformulation of the original system in the form of a system of conservation laws coupled to a Hamilton–Jacobi equation for each phase boundary. The numerical method for the reformulated system is constructed for unstructured meshes (in 2D), and a self-adaptive algorithm is introduced.In the 1D-case we show that the reformulated system has a solution that corresponds to exact dynamical phase boundaries of the elasticity system which obey the kinetic relation. To validate the method in 2D, we present computations on the interaction of a plane wave with a phase boundary. The efficiency of the adaptation mechanism is demonstrated by an example showing the development of microstructures by twinning.     

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016     

Conservative finite difference schemes for the generalized Zakharov–Kuznetsov equations

This paper is concerned with the construction of conservative finite difference schemes by means of discrete variational method for the generalized Zakharov–Kuznetsov equations and the numerical solvability of the two-dimensional nonlinear wave equations. A finite difference scheme is proposed such that mass and energy conservation laws associated with the generalized Zakharov–Kuznetsov equations hold. Our arguments are based on the procedure that D. Furihata has recently developed for real-valued nonlinear partial differential equations. Numerical results are given to confirm the accuracy as well as validity of the numerical solutions and then exhibit remarkable nonlinear phenomena of the interaction and behavior of pulse wave solutions.