A self-optimizing single-step algorithm for the numerical integration of parabolic equations Original Research Article

An unconditionally stable single step time marching algorithm for parabolic equations is presented. In this algorithm, time stepping parameters are chosen according to an error condition such that large time step accuracy is achieved, and the solution is also free from oscillations. It is different from conventional algorithms, such as the Crank-Nicolson scheme, in that it uses a matrix of time stepping parameters rather than a single scalar value. These parameters can be approximated inexpensively. This algorithm is easily programmed and comparisons are made with other methods for different cases of a heat conduction problem. It is found that the algorithm gives a more accurate solution for all degrees-of-freedom for the cases considered. It is therefore advantageous over the conventional variably weighted Euler θ-method which has a constraint that the same θ value must be used for all degrees-of-freedom and that θ values have to be changed by trial and error for different time step sizes and mesh refinements in order to achieve best results.