Solving stiff differential equations with the method of patches
A new approach for solving stiff systems of nonlinear ordinary differential equations is presented. In the case of an autonomous scalar differential equation \( \dot x (t) = f (x(t)) \), the authors propose to define a suitable grid in the \(x\)-variable \( x_0 < x_1 < \ldots x_N\) replacing the original system by \( \dot y (t) = f_L ( y(t))\) where \( f_L(y)\) is a piecewise linear function which interpolates \( f(x)\) on each interval \([x_i, x_{i+1}]\). Now the solution of the new piecewise linear equation can be expressed in terms of elementary functions and they obtain an approximate solution to the original problem. For two-dimensional autonomous systems \( \dot x = f(x)\) the authors propose to use triangular elements to interpolate the nonlinear function \( f\) on the corners of each triangle. Further they pay attention to choose appropriately the nodes so that the linearized equations retain the main features of the original flow. Although the idea of the new method is very simple, its practical implementation has some difficulties already pointed out by the authors in a couple of examples. Among them we mention the determination of the time arrival to the boundary of the triangular element and the reliable calculation of the exponential of a matrix. The paper includes also additional studies on the accuracy of the solution provided by the method and the effect of a least squares fitting of \( f \) instead of a linear interpolation on the accuracy of the method.
scientific article; zbMATH DE number 1282689
scientific article