A fully adaptive MOL-treatment of parabolic 1-D problems with extrapolation techniques

Abstract A fully adaptive method is presented for the numerical solution of highly nonlinear, coupled systems of parabolic differential equations in one space dimension. Time discretization is by means of the linearly implicit Euler discretization. Space discretization is by finite differences on nonuniform grids. Both basic discretizations are combined with extrapolation. Based on local error estimates for both the time and the space discretization error, the accuracy of the numerical approximation is controlled and the discretization stepsizes are adapted automatically and simultaneously. The algorithm is implemented in a user friendly software package, PDEX1M.

[1]  C. Lubich,et al.  Linearly implicit extrapolation methods for differential-algebraic systems , 1989 .

[2]  Keith Miller,et al.  Moving Finite Elements. I , 1981 .

[3]  Eugenia Kálnay de Rivas.  On the use of nonuniform grids in finite-difference equations , 1972 .

[4]  P. Deuflhard,et al.  Extrapolation Integrators for Quasilinear Implicit ODEs , 1987 .

[5]  E. Dorfi,et al.  Simple adaptive grids for 1-d initial value problems , 1987 .

[6]  Richard F. Sincovec,et al.  Software for Nonlinear Partial Differential Equations , 1975, TOMS.

[7]  L. J. Comrie,et al.  Recent Progress in Scientific Computing , 1944 .

[8]  Peter Deuflhard,et al.  Recent progress in extrapolation methods for ordinary differential equations , 1985 .

[9]  James M. Hyman.  MOVING MESH METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS , 1988 .

[10]  E. Hairer,et al.  Dense output for extrapolation methods , 1990 .

[11]  P. Deuflhard.  Order and stepsize control in extrapolation methods , 1983 .

[12]  Joseph E. Flaherty,et al.  A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations , 1986 .

[13]  Linda R. Petzold,et al.  Observations on an adaptive moving grid method for one-dimensional systems of partial differential equations , 1987 .

[14]  Alexander Ostermann,et al.  RUNGE-KUTTA METHODS FOR PARABOLIC EQUATIONS AND CONVOLUTION QUADRATURE , 1993 .

[15]  Joseph E. Flaherty,et al.  Integrated space-time adaptive hp -refinement methods for parabolic systems , 1995 .