On the derivation of the modified equation for the analysis of linear numerical methods
On the discrete maximum principle for parabolic difference operators
On the Estimates of the Convergence Rate of the Finite Difference Schemes for the Approximation of Solutions of Hyperbolic Problems
On the Estimates of the Convergence Rate of the Finite Difference Schemes for the Approximation of Solutions of Hyperbolic Problems, II Part
On the Existence of Generalized Solutions and the Convergence of Difference Methods for Nonlinear Initial-Value Problems.
On the numerical approximation of first-order Hamilton-Jacobi equations
Some methods for the numerical approximation of time-dependent and steady first-order Hamilton-Jacobi equations are reviewed. Most of the discussion focuses on conformal triangular-type meshes, but we show how to extend this to the most general meshes. We review some first-order monotone schemes and also high-order ones specially dedicated to steady problems.
On the numerical solution of fractional hyperbolic partial differential equations.
On the numerical solution of the one-dimensional convection-diffusion equation.
On the parallel domain decomposition algorithms for time-dependent problems.
On the stability of the linear delay differential and difference equations.
On the strong stability of operator-difference schemes in time-integral norms.
On the Structure of Error Estimates for Finite-Difference Methods.
On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems
This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys. 102 (1992) 360–373]. Next, this general theory is applied to obtain well-balanced...
On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems
This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys.102 (1992) 360–373]. Next, this general theory is applied to obtain well-balanced...
On well-posedness of the nonlocal boundary value problem for parabolic difference equations.
Operator-splitting schemes for solving elasticity problems.
Opposing flows in a one dimensional convection-diffusion problem
In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference operator...
Optimal grids for anisotropic problems.
Parabolic-elliptic correspondence of a three-level finite difference approximation to the heat equation.
Parallel algorithm for solving the Black-Scholes equation