Estimating the Spectral Radius of the Jacobian Matrix in Explicit Stabilized Runge–kutta Methods

When using explicit stabilized Runge–Kutta methods to solve stiff systems of ordinary differential equations, an estimate of the spectral radius of the Jacobian matrix is required. Such an estimate can be obtained by applying Gershgorins theorem or the power method. This paper investigates estimation procedures based on the nonlinear power method that do not require computation of the Jacobian matrix. The proposed procedures are embedded in the integration method and allow estimating the spectral radius even when it changes during the solution process. Examples of solving test problems are provided.