Runge Kutta Chebyshev Method for parabolic ... - Brown University

Introduction Properties Pros and Cons Examples References
Introduction
initial value problem for the ODE systems:
˙u(t) = F(t, u(t)), 0 < t � T, u(0) = u0
which originate from spatial discretization of parabolic PDEs.
Restrictions:
The eigenvalues of the Jacobian matrix should lie in a
narrow strip along the negative axis of the complex plane
Jacobian matrix should not deviate too much from a
normal matrix.
Example:
model heat equation
reaction-diffusion problem
(1)
˙u(t) = ∆u (2)
˙u(t) = ɛ∆u + f(u, x, t), 0 < ɛ ≪ 1 (3)