In the last decade, several numerical techniques have been developed to solve time-dependent partial differential equations (PDEs) in one dimension having solutions with steep gradients in space and in time. One of these techniques, a moving-grid method based on a Lagrangian description of the PDE and a smoothed-equidistribution principle to define the grid positions at each time level, has been coupled with a spatial discretization method that automatically discreizes the spatial part of the user-defined PDE following the method of lines approach. We supply two FORTRAN subroutines, CWRESU and CWRESX, which compute the residuals of the differential algebraic equations (DAE) system obtained from semidiscretizing, respectively, the PDE and the set of moving-grid equations. These routines are combined in an enveloping routine SKMRES, which delivers the residuals of the complete DAE system. To solve this stiff, nonlinear DAE system, a robust and efficient time-integrator must be applied, for example, a BDF method such as implemented in the DAE solvers SPRINT [Berzins and Furzeland 1985; 1986; Berzins et al. 1989] and DASSL [Brenan et al. 1989; Petzold 1983]. Some numerical examples are shown to illustrate the simple and effective use of this software interface.

Algorithm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations

An error controlled finite element method (FEM) for solving stationary Schrodinger equations in three space dimensions is proposed. The method is based on an adaptive space discretization into tetrahedra and local polynomial basis functions of order p=1–5 defined on these tetrahedra. According to a local error estimator, the triangulation is automatically adapted to the solution. Numerical results for standard problems appearing in vibrational motion and molecular structure calculations are presented and discussed. Relative precisions better than 1e‐8 are obtained. For equilateral H++3, the adaptive FEM turns out to be superior to global basis set expansions in the literature. Our precise FEM results exclude in a definite manner the stability or metastability of equilateral H++3 in its ground state.

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A self-adaptive multilevel finite element method for the stationary Schrödinger equation in three space dimensions

Large-scale combustion simulations show the need for adaptive methods. First, to save computation time and mainly to resolve local and instationary phenomena. In contrast to the widespread method of lines, we look at the reaction-diffusion equations as an abstract Cauchy problem in an appropriate Hilbert space. This means we first discretize in time, assuming the space problems solved up to a prescribed tolerance. So, we are able to control the space and time error separately in an adaptive approach. The time discretization is done by several adaptive Runge-Kutta methods, whereas for the space discretization a finite element method is used. The different behavior of the proposed approaches is demonstrated on many fundamental examples from ecology, flame propagation, electrodynamics, and combustion theory.

A finite element method adaptive in space and time for nonlinear reaction-diffusion systems

We consider an approach to determine parameters in a system of one-dimensional time-dependent parabolic differential equations and coupled ordinary differential equations. The model allows transmis...

http://www.klaus-schittkowski.de/reportPDE1D.pdf

Miente Bakker

Papers

Software for semi-discretization of time-dependent partial differential equations in one space variable : (preprint)