A predictor-corrector method for a certain class of stiff differential equations

Abstract

Abstract.In stiff systems of linear ordinary differential equations, certain elements of the matrix describing the system are very large. Sometimes, e.g., in treating partial differential equations, the problem can be formulated in such a manner that large elements appear only in the main diagonal. Then the elements causing stiffness can be taken into account analytically. This is the basis of the predictor-corrector method presented here. The truncation error can be estimated in terms of the difference between predicted and corrected values in nearly the same manner as for the customary predictor-corrector method. The question of stability, which is crucial for stiff equations, is first studied for a single equation; as expected, the method is much more stable than the usual predictor-corrector method. For systems of equations, sufficient conditions for stability are derived which require less work than a detailed stability analysis. The main tool is a matrix norm which is consistent with a weighted infinity vector norm. The choice of the weights is critical. Their determination leads to the question whether a certain matrix has a positive inverse.

An exponential method for the solution of systems of ordinary differential equations

“…(5) is similar to the starting point for several other authors, e.g. Certaine [6], Pope [7J, Guderley and Hsu [8], and Sarkany and Liniger [12]. In [6 and 8], the matrix A of (1) is assumed to be constant and diagonal throughout the interval [0,a]; the function F is then interpolated by polynomials resulting in multistep formulas.…”

“…This is in fact done in the actual computer program. An initial difference formula, in component form is (8) aij A'/ + i^,'/shk)Gih +…”

“…Since The difference formula ((8) or (8')) can be further improved by means of a correction term. This is accomplished by taking into consideration the difference between the solution as predicted by (8) and the initial extrapolation. Let AX,' = ^X'^' -X' -h^X', and consider the new extrapolations:…”

Formal Solutions for Polarized Radiative Transfer. I. The DELO Family

“…The first method of this family applied to the formal solution for polarized light was proposed by Rees et al (1989) As exhaustively explained by Guderley & Hsu (1972), this technique takes into account analytically the diagonal elements of the propagation matrix K, aiming to remove stiffness from the problem. Therefore, the well-known radiative transfer equation for polarized light given by Equation (1) is brought in the form given by Equation (12), with the additional constraint of a diagonal matrix L. This reformulation is facilitated by the fact that the diagonal elements of the propagation matrix are all identical.…”

“…The integral can then be solved by parts, yielding an implicit or explicit linear system for the Stokes vector I k+1 . As explained by Guderley & Hsu (1972), the local truncation error is due to the fact that the effective source function is approximated by a polynomial of degree q. According to Henrici (1962), this approximation results in the following local truncation error…”