We describe the development of a flux-limited gray radiation solver for the compressible astrophysics code, CASTRO. CASTRO uses an Eulerian grid with block-structured adaptive mesh refinement based on a nested hierarchy of logically-rectangular variable-sized grids with simultaneous refinement in both space and time. The gray radiation solver is based on a mixed-frame formulation of radiation h...

Abstract. Variational steepest descent approximation schemes for the modified Patlak-KellerSegel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recov...

The paper deals with discretisation methods for nonlinear operator equations written as abstract nonlinear evolution equations. Brezis and Pazy showed that the solution of such problems is given by nonlinear semigroups whose theory was founded by Crandall and Liggett. By using the approximation theorem of Brezis and Pazy, we show the N-stability of the abstract nonlinear discrete problem for th...

Abstract. Partitioned systems of ordinary differential equations are in qualitative terms characterized as monotonically max-norm stable if each sub-system is stable and if the couplings from one sub-system to the others are weak. Each sub-system of the partitioned system may be discretized independently by the backward Euler formula using solution values from the other sub-systems correspondin...

In this paper we describe the classical methods used to solve the Euler equations. Special attention is devoted to the iterative method based on a contraction mapping derived from these equations in Maldonado and Moreira (2003). We test the numerical robustness of this method when it is used in models with sensitiveness to initial conditions. Finally we extend the method to the case of stochast...

To avoid finding the stationary distributions of stochastic differential equations by solving the nontrivial Kolmogorov-Fokker-Planck equations, the numerical stationary distributions are used as the approximations instead. This paper is devoted to approximate the stationary distribution of the underlying equation by the Backward Euler-Maruyama method. Currently existing results [21, 31, 33] ar...

A rather general semilinear parabolic problem is studied together with its spatially semidiscrete nite element approximation. Both problems are formulated within the framework of nonlinear semigroups in the Sobolev space H 1 ((). The main result is an error estimate for solutions with initial data in H 1 ((), valid during an arbitrary nite time interval. The proof is based on the semigroup form...

We consider numerical approximation of solutions of singular second order differential equations. In particular, we study the backward (or implicit) Euler method. We prove results concerning consistency, global error and stability. We show that the global error is linear with respect to the step size. Numerical results are also given, which demonstrate the linear convergence and compare the num...

The importance of stochastic differential delay equations (SDDEs) derives from the fact that many of the phenomena witnessed around us do not have an immediate effect from the moment of their occurrence. A patient, for example, shows symptoms of an illness days (or even weeks) after the day in which he or she was infected. In general, we can find many ”systems”, in almost any area of science (m...

In this paper we describe the classical methods used to solve the Euler equations. Special attention is devoted to the iterative method based on a contraction mapping derived from these equations in Maldonado and Moreira (2003). We test the numerical robustness of this method when it is used in models with sensitiveness to initial conditions. Finally we extend the method to the case of stochast...