Region-dependent optimal \(m\)-stage Runge-Kutta schemes for solving a class of nonsymmetric linear systems
Consider the system of linear algebraic equations \(Ax= b\), where \(A\) is an \(N\times N\) nonsymmetric real matrix with the following properties: (a) The matrix \(A\) is nonsingular and has a complete eigenspace, (b) The real parts of all eigenvalues are nonnegative. The paper presents some analytic and numerical results on \(m\)-stage Runge-Kutta schemes for solving the above system. The solution of the algebraic system is treated as the steady state of a system of ordinary differential equations, \(dx/dy+ Ax= b\). For each \(m\), the authors find a compact complex region \(\Omega\) which has a simply connected complement and contains all eigenvalues of the system. A specially structured nonlinear minimax problem over the region is formulated to find the optimal parameters of the Runge-Kutta method.
system of ordinary differential equations
scientific article; zbMATH DE number 720494
scientific article