Time-step sequences for parabolic differential equations (Q1893548)

The Euler forward difference method is used for the explicit time integration of stiff systems of ordinary differential equations, which originate from spatial discretization of parabolic partial differential equations. The author constructs a sequence, with which the integration proceeds faster and faster in time by taking bigger and bigger steps, but by which the precision of the approximation is guaranteed throughout the integration procedure. This sequence is primarily a function of one given parameter which determines an upper bound for the distance in norm between the analytical solution and the approximation at every discrete time level in case the Jacobian matrix is constant. This method makes use of Chebyshev polynomials. The asymptotic rate of convergence of the method is derived and its applicability in case the Jacobian matrix is non-constant is demonstrated.