An Error Corrected Euler Method for Solving Stiff Problems Based on Chebyshev Collocation

In this paper, we present error corrected Euler methods for solving stiff initial value problems, which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess. The proposed methods use a Chebyshev collocation technique as well as an asymptotical linear ordinary differential equation of first-order derived from the difference between the exact solution and the Euler’s polygon. These methods with or without the Jacobian are analyzed in terms of convergence and stability. In particular, it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the Jacobian. Numerical tests are given to support the theoretical analysis as evidences.