Locally One-dimensional Research Articles

New lod and adi methods for the two-dimensional diffusion equation

This paper extends earlier work on the solution of the constant-coefficient two-dimensional diffusion equation by considering two classes of time-split finite-difference methods, namely locally one-dimensional (LOD) schemes and alternating direction implicit (ADI) methods. Two new fourth-order techniques are described and tested. Firstly, a LOD method based on the fourth-order explicit Noye-Hayman procedure for the one-dimensional diffusion equation is described. Proper treatment of values at, and adjacent to, the boundary at intermediate time levels is necessary, otherwise the method degenerates to second-order. Secondly, an unconditionally stable ADI method based on a (3,9) two-dimensional computational molecule is developed and tested.

Unconditional convergence of some Crank-Nicolson LOD methods for initial-boundary value problems

In this paper convergence properties are discussed for some locally one-dimensional (LOD) splitting methods applied to linear parabolic initial-boundary value problems. We shall consider unconditional convergence, where both the stepsize in time and the meshwidth in space tend to zero, independently of each other.

Unconditional Convergence of Some Crank-Nicolson Lod Methods for Initial- Boundary Value Problems

In this paper convergence properties are discussed for some locally one-dimensional (LOD) splitting methods applied to linear parabolic initialboundary value problems. We shall consider unconditional convergence, where both the stepsize in time and the meshwidth in space tend to zero, independently of each other.

Comparison of two splitting algorithms for 2-D modelling of advection-dominated solute transport in groundwater

This paper reports on two alternating directions splitting algorithms, one cast in the Local One Dimensional (LOD) and the other in the standard Fractional Time Stepping (FTS) format. Both algorithms employ an adaptation of the pseudo-viscosity technique and use the principal directions of transport as coordinates. In the LOD approach, only one dimension appears in each splitting step. In the FTS method, computation is implicit in only one dimension but other dimensions are present in each splitting step. Stability and accuracy constraints on the grid parameters are diagrammed for both schemes. Test problems in one and two dimensions are solved. The analysis shows that the slight accuracy advantage of the FTS format over the LOD format must be traded off against tighter grid design prescriptions and, frequently, also against higher computing cost.

Time-Dependent Boundary and Interior Forcing in Locally One-Dimensional Schemes

This paper is concerned with the numerical solution of parabolic partial differential equations with Dirichlet boundary conditions by locally one-dimensional (LOD) methods. When a corresponding system of ordinary differential equations is developed for the numerical time-integration, the prescribed values of the solution (boundary values) and any interior forcing (resulting either from local linearization or from interior source terms in the original equations) must be treated differently to prevent loss in the order of accuracy of the numerical solution as compared to a homogeneous (unforced) problem. A stable second-order scheme and a global extrapolation to third order are demonstrated.

The construction of hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative

Hopscotch, a fast finite difference technique, is used to solve parabolic and elliptic equations in two space dimensions with a mixed derivative. The method is compared numerically with existing alternating direction implicit (A.D.I.) and locally one dimensional (L.O.D.) methods for simple problems. Douglas and Gunn's A.D.I. method is both simplified and improved by reformulating it as a hopscotch method.

On the Structure of Alternating Direction Implicit (A.D.I.) and Locally One Dimensional (L.O.D.) Difference Methods

On the Structure of Alternating Direction Implicit (A.D.I.) and Locally One Dimensional (L.O.D.) Difference Methods