A method for the spatial discretization of parabolic equations in one space variable. (English) Zbl 0701.65065

Skeel, Robert D.; Berzins, Martin

SIAM J. Sci. Stat. Comput. 11, No. 1, 1-32 (1990).

Let us consider the system of n quasilinear partial differential equations \[ (1)\quad D(x,t,u,u_ x)u_ t=x^{-m}(x^ mg(x,t,u,u_ x))_ x+f(x,t,u,u_ x) \] for \(a\leq x\leq b\) where D is a diagonal matrix with nonnegative entries, \(m\geq 0\) and boundary conditions \(p^ i(x,t,u)+g^ i(x,t)g^ i(x,t,u,u_ x)=0\quad at\quad x=a,b\) for \(i=1,2,...,n\). The authors choose the problem class defined by (1) so as to have recognizable flux and source terms and to have the possibility of recognizable Cartesian polar and spherical polar coordinates.
The paper is concerned with the design of a spatial discretization method for polar and nonpolar parabolic equations (1). A new spatial discretization method suitable for use in a library program is derived. It is a simple piecewise nonlinear Galerkin/Petrov-Galerkin method that is second-order accurate in space. The case \(m=1\) involves the use of the logarithm function which enables us to model precisely the logarithmic behaviour that can be present in the solution.
The developed method is compared with the package SPRINT and several other known packages. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithm.
Theoretical and experimental evidence indicates that the derived Galerkin method for the regular case and the Petrov-Galerkin method for the singular case produce more accurate results than existing methods to use in a general purpose library subroutine.

Reviewer: J.Hřebiček

MSC:

65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
35K55	Nonlinear parabolic equations
35-04	Software, source code, etc. for problems pertaining to partial differential equations

Keywords:

singular boundary value problem; method of lines; quasilinear; polar and nonpolar parabolic equations; Truncation error; numerical examples; algorithm; Galerkin method; Petrov-Galerkin method; library subroutine

Software:

SPRINT2D