The multiscale multigrid method is presented in this article to solve the linear systems arising from a fourth order discretisation. We used a symbolic algebra packageMathematica to derive a family of finite difference approximations on a 27 point compact stencil. The unknown solution and its second derivatives are carried as unknowns at selected grid points. A set of test problems are presente...

This paper discusses compact-stencil nite difference time domain (FDTD) schemes for approximating the 2D wave equation in the context of digital audio. Stability, accuracy, and ef ciency are investigated and new ways of viewing and interpreting the results are discussed. It is shown that if a tight accuracy constraint is applied, implicit schemes outperform explicit schemes. The paper also disc...

In this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting, alternating direction or, more generally, fractional step methods. Such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive Runge–Kutta method, which is called the fractional step Runge–Kutta met...

In this article we discuss a combination between fourth-order finite difference methods and fourth-order splitting methods for 2D parabolic problems with mixed derivatives. The finite difference methods are based on higher-order spatial discretization methods, whereas the timediscretization methods are higher-order discretizations using CrankNicolson or BDF methods. The splitting methods are hi...