Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

Summary. The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges strongly in in a finite volume sense. Received February 25, 1997 / Revised version received November 8, 1999 / Published online August 24, 2000     

Convergence rates to discrete shocks for nonconvex conservation laws

Summary. This paper is concerned with polynomial decay rates of perturbations to stationary discrete shocks for the Lax-Friedrichs scheme approximating non-convex scalar conservation laws. We assume that the discrete initial data tend to constant states as , respectively, and that the Riemann problem for the corresponding hyperbolic equation admits a stationary shock wave. If the summation of the initial perturbation over is small and decays with an algebraic rate as , then the perturbations to discrete shocks are shown to decay with the corresponding rate as . The proof is given by applying weighted energy estimates. A discrete weight function, which depends on the space-time variables for the decay rate and the state of the discrete shocks in order to treat the non-convexity, plays a crucial role. Received November 25, 1998 / Published online November 8, 2000     

Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains

Summary. This paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains. General assumptions are made on the data: the initial condition and the boundary condition are supposed to be measurable bounded functions. Using a generalized notion of solution to the continuous problem (namely the notion of entropy process solution, see [9]) and a uniqueness result on this solution, we prove that the numerical solution converges to the entropy weak solution of the continuous problem in for every . This also yields a new proof of the existence of an entropy weak solution. Received May 18, 2000 / Revised version received November 21, 2000 / Published online June 7, 2001     

Generalized multiresolution analysis on unstructured grids

Summary. Efficiency of high-order essentially non-oscillatory (ENO) approximations of conservation laws can be drastically improved if ideas of multiresolution analysis are taken into account. These methods of data compression not only reduce the necessary amount of discrete data but can also serve as tools in detecting local low-dimensional features in the numerical solution. We describe the mathematical background of the generalized multiresolution analysis as developed by Abgrall and Harten in [14], [15] and [3]. We were able to ultimately reduce the functional analytic background to matrix-vector operations of linear algebra. We consider the example of interpolation on the line as well as the important case of multiresolution analysis of cell average data which is used in finite volume approximations. In contrast to Abgrall and Harten, we develop a robust agglomeration procedure and recovery algorithms based on least-squeare polynomials. The efficiency of our algorithms is documented by means of several examples. Received April 4, 1998 / Revised version August 2, 1999 / Published online June 8, 2000     

An entropy satisfying MUSCL scheme for systems of conservation laws

Summary. For the high-order numerical approximation of hyperbolic systems of conservation laws, we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by Harten for scalar equations. Based on this new criterion, we derive entropy diminishing projections that ensure, both, the second order of accuracy and all of the classical discrete entropy inequalities. The resulting scheme is a nonlinear version of the classical Van Leer's MUSCL scheme. Strong convergence of this second order, entropy satisfying scheme is proved for systems of two equations. Numerical tests demonstrate the interest of our theory. Received March 28, 1995 / Revised version received June 17, 1995     

Third order nonoscillatory central scheme for hyperbolic conservation laws

Summary. A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: 1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given cell averages; and 2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a third-order central scheme, an extension along the lines of the second-order central scheme of Nessyahu and Tadmor \cite{NT}. The scalar scheme is non-oscillatory (and hence – convergent), in the sense that it does not increase the number of initial extrema (– as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the high-resolution content of the proposed scheme. Thus, a considerable amount of simplicity and robustness is gained while retaining the expected third-order resolution. Received April 10, 1996 / Revised version received January 20, 1997     

Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions

Summary. We prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. The result is applied to the discontinuous Galerkin method due to Cockburn, Hou and Shu. Received April 15, 1993 / Revised version received March 13, 1995     

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

Summary. One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation by a piecewise constant function using a discretization in space and time and a finite volume scheme. The convergence of to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on are used to prove the convergence, up to a subsequence, of to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of to{\it u}. Some on a model equation are shown. Received September 27, 2000 / Published online October 17, 2001     

A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

Summary. We construct a new third-order semi-discrete genuinely multidimensional central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of , allows us to pass to a limit as . This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme. Received February 7, 2000 / Published online December 19, 2000     

Convergence analysis and error estimates for a parallel algorithm for solving the Navier-Stokes equations

Summary. This paper is concerned with the analysis of the convergence and the derivation of error estimates for a parallel algorithm which is used to solve the incompressible Navier-Stokes equations. As usual, the main idea is to split the main differential operator; this allows to consider independently the two main difficulties, namely nonlinearity and incompressibility. The results justify the observed accuracy of related numerical results. Received April 20, 2001 / Revised version received May 21, 2001 / Published online March 8, 2002 RID="*" ID="*" Partially supported by D.G.E.S. (Spain), Proyecto PB98–1134 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="*" ID="*" Partially supported by D.G.E.S. (Spain), Proyecto PB98–1134 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="**" ID="**" Partially supported by D.G.E.S. (Spain) Proyecto PB96–0986     

Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations

Summary. This paper considers the questions of convergence of: (i) MUSCL type (i.e. second-order, TVD) finite-difference approximations towards the entropic weak solution of scalar, one-dimensional conservation laws with strictly convex flux and (ii) higher-order schemes (filtered to ``preserve' an upper-bound on some weak second-order finite differences) towards the viscosity solution of scalar, multi-dimensional Hamilton-Jacobi equations with convex Hamiltonians. Received May 16, 1994     

Error bounds for the large time step Glimm scheme applied to scalar conservation laws

Summary. In this paper we derive an error bound for the large time step, i.e. large Courant number, version of the Glimm scheme when used for the approximation of solutions to a genuinely nonlinear, i.e. convex, scalar conservation law for a generic class of piecewise constant data. We show that the error is bounded by for Courant numbers up to 1. The order of the error is the same as that given by Hoff and Smoller [5] in 1985 for the Glimm scheme under the restriction of Courant numbers up to 1/2. Received April 10, 2000 / Revised version received January 16, 2001 / Published online September 19, 2001     

Convergence of a relaxation scheme for hyperbolic systems of conservation laws

Summary. This paper concerns the study of a relaxation scheme for hyperbolic systems of conservation laws. In particular, with the compensated compactness techniques, we prove a rigorous result of convergence of the approximate solutions toward an entropy solution of the equilibrium system, as the relaxation time and the mesh size tend to zero. Received September 29, 1998 / Revised version received December 20, 1999 / Published online August 24, 2000     

Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Summary. In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law with initial condition . The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is in space-time -norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case. Received October 21, 1999 / Published online February 5, 2001     

A priori error estimates for approximate solutions to convex conservation laws

Summary. We introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite–difference approximation calculated with the scheme of Engquist-Osher, Lax-Friedrichs, or Godunov. This technique is a discrete counterpart of the duality technique introduced by Tadmor [SIAM J. Numer. Anal. 1991]. The error is related to the consistency error of cell averages of the entropy weak solution. This consistency error can be estimated by exploiting a regularity structure of the entropy weak solution. One ends up with optimal error estimates. Received December 21, 2001 / Revised version received February 18, 2002 / Published online June 17, 2002     

A fully discrete scheme for diffusive-dispersive conservation laws

Summary.   We introduce a fully discrete (in both space and time) scheme for the numerical approximation of diffusive-dispersive hyperbolic conservation laws in one-space dimension. This scheme extends an approach by LeFloch and Rohde [4]: it satisfies a cell entropy inequality and, as a consequence, the space integral of the entropy is a decreasing function of time. This is an important stability property, shared by the continuous model as well. Following Hayes and LeFloch [2], we show that the limiting solutions generated by the scheme need not coincide with the classical Oleinik-Kruzkov entropy solutions, but contain nonclassical undercompressive shock waves. Investigating the properties of the scheme, we stress various similarities and differences between the continuous model and the discrete scheme (dynamics of nonclassical shocks, nucleation, etc). Received November 15, 1999 / Revised version received May 27, 2000 / Published online March 20, 2001     

Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation

Summary. When numerically integrating time-dependent differential equations, it is often recommended to employ methods that preserve some of the invariant quantities (mass, energy, etc.) of the problem being considered. This recommendation is usually justified on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative features. However there are cases where schemes that preserve invariants are also advantageous in that they possess favourable error propagation mechanisms that render them superior from a quantitative point of view. In the present paper we consider the Korteweg-de Vries equation as a case study. We show rigorously that, for soliton problems and at leading order, the error of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded in the norm uniformly in time. For ‘general’, nonconservative schemes the error involves a linearly growing amplitude error, a quadratically growing phase error and a complementary term that grows linearly in the norm. Numerical experiments are presented. Received November 21, 1994 / Revised version received July 17, 1995     

Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids

Summary. We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption on the substructures that constitute the whole domain, so the substructures can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved, and neither the coarse mesh nor the fine mesh need be quasi-uniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate as the usual two level overlapping domain decomposition methods on structured meshes. The condition number of the preconditioned system depends only on the (possibly small) overlap of the substructures and the size of the coarse grid, but is independent of the sizes of the subdomains. Received March 23, 1994 / Revised version received June 2, 1995     

Convergence of finite volume schemes for semilinear convection diffusion equations

The topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered finite volume method. Dirichlet boundary conditions are considered here. A discrete Poincaré inequality is used, and estimates on the approximate solutions are proven. The convergence of the scheme without any assumption on the regularity of the exact solution is proven using some compactness results which are shown to hold for the approximate solutions. Received January 16, 1998 / Revised version received June 19, 1998     

Convergence of Runge–Kutta approximations for parabolic problems with Neumann boundary conditions

Summary. We analyze a class of algebraically stable Runge–Kutta/standard Galerkin methods for inhomogeneous linear parabolic equations, with time–dependent coefficients, under Neumann boundary conditions, and derive an error bound of provided is bounded. Received June 25, 1994 / Revised version received February 26, 1996