Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations
For the one-dimensional heat equation with or without a linear functional term (covering e.g. linear integro-differential equations or equations with a time delay), the numerical solution by means of the waveform relaxation is studied. After the spatial discretization, which uses finite differences of second-order or a pseudospectral method based upon Chebyshev polynomials of the first kind, the Gauß-Seidel waveform relaxation technique is applied. This results in an iterative sequence of systems of ordinary differential equations, each described by a lower triangle matrix. Hence, a solution of the equations from up to down by e.g. the backward differentiation formulae (BDF), as proposed by the author, does not require the solution of an algebraic system. For both, the waveform relaxation after the finite difference method and after the Chebyshev method, resp., the time-dependent error is analyzed by means of estimates using logarithmic norms. Numerical examples illustrate the two different methods. Here, for the time discretization, the BDF of order 3 with constant step size has been used. Better convergence and computational efficiency has been observed when using the Chebyshev method instead of the method of finite differences.
differential-functional parabolic equations
scientific article; zbMATH DE number 1462306