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 parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which ...

A singularly perturbed boundary-value problem with a multiple turning point at a boundary is considered. A representation of the solution is given, and it is used in the construction of a uniform finite-difference scheme. The scheme is a fiNJt-order exponentially fitted one. An improved modification on a special discretization mesh is given.

In this paper we study the heat and advection equation in single and multiple domains. We discretize using a second order accurate finite difference method on Summation-By-Parts form with weak boundary and interface conditions. We derive analytic expressions for the spectrum of the continuous problem and for their corresponding discretization matrices. We show how the spectrum of the single dom...

The initial-boundary value problem for a linear parabolic equation with the Dirichlet boundary condition is solved approximately by applying the finite element discretization in the space dimension and three types of finite-difference discretizations in time: the backward, the Crank-Nicolson and the Calahan discretization. New error bounds are derived.

where the residual vector R contains high-order derivatives of the dependent variable u(t, x1, x2, . . . , xn) with respect to independent variables. Equation (1) is an abstract representation of a generic Partial Differential Equation (PDE) in the residual form where the linear operator L accounts for a linear combination of the k partial derivative in the i direction, i.e. ∂/∂xi . When discre...

We investigate the limiting description for a finite-difference approximation of a singularly perturbed Allen–Cahn type energy functional. The key issue is to understand the interaction between two small length-scales: the interfacial thickness ε and the mesh size of spatial discretization δ. Depending on their relative sizes, we obtain results in the framework of Γ-convergence for the (i) subc...

We consider hyperbolic systems of conservation laws which are discretized in space by spectral collocation methods and advanced in time by finite difference schemes. At any time-level we introduce a domain decomposition method based on an iteration-by-subdomain procedure yielding at each step a sequence of independent subproblems (one for each subdomain) that can be solved simultaneously. The m...

In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain Ω× [0, T ] which is decomposed into an overlapping collection of cylindrical subregions of the form Ωl ×[0, T ], fo...

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are not necessarily equal. The components of the solution exhibit overlapping layers. A Shishkin piecewise– uniform mesh is constructed, which is used, in conjunction with a cl...