Two Methods for the Implicit Integration of Stiff Reaction Systems

Abstract

Abstract We present two methods for the implicit integration of nonlinear stiff systems. Direct application of the Newton method to backward Euler discretization of such systems may diverge. We observe that the solution is recovered by smoothing out certain eigenvalues in the Jacobian matrix. To this end, we introduce a solution-dependent matrix-weighted combination of backward and forward Euler methods. The weight is tuned on each Newton iteration to reproduce the solution with an exponential integrator, whereby a weight function for smoothing eigenvalues is obtained. We apply the proposed techniques, namely quasi-Newton backward Euler and matrix-weighted Euler, to several stiff systems, including Lotka–Volterra, Van der Pol’s, and a blood coagulation cascade.