Stabilized Explicit Runge-Kutta Methods

Abstract

In this chapter we discuss special purpose explicit Runge-Kutta methods for systems of ODEs in ℝ^m

$$ w'\left( t \right) = F\left( {t,w\left( t \right)} \right),\quad t >0,\quad w\left( 0 \right) = {w_0},$$

representing semi-discrete, multi-space dimensional parabolic problems. Often, parabolic problems give rise to stiff systems having a symmetric Jacobian matrix ∂F(t,w)/∂wwith a spectral radius proportional to h ^-2, hrepresenting a spatial mesh width. Standard explicit methods are then highly inefficient due to their severe stability constraint, see Section II.1.4. On the other hand, unconditionally stable implicit methods, like backward Euler or the implicit trapezoidal rule, do require one or more linear or nonlinear algebraic system solutions at each integration step, which can become costly in higher space dimension.