336 research outputs found
Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear reaction-diffusion system
A singularly perturbed linear system of second order ordinary differential
equations of reaction-diffusion type with given boundary conditions is
considered. The leading term of each equation is multiplied by a small positive
parameter. These singular perturbation parameters are assumed to be distinct.
The components of the solution exhibit overlapping layers. Shishkin
piecewise-uniform meshes are introduced, which are used in conjunction with a
classical finite difference discretisation, to construct a numerical method for
solving this problem. It is proved that the numerical approximations obtained
with this method is essentially second order convergent uniformly with respect
to all of the parameters
Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type
In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques
A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems
In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm
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