Collocation and Galerkin Time-Stepping Methods

Abstract

[Abstract] We study the numerical solutions of ordinary differential equations by one -step method s where t he solution at n t is known and that at 1 + n t is to be calculated . The approache s employed are col location, continuous Galerkin (C G) and discontinuous Galerkin (D G). Relations among these three approaches are establish ed . A quadrature formula using s evaluation points is employed for the Galerkin formulation s. We show that with such a quadrature , the CG method is identical to the collocation method using quadrature points as collocation points. Fur thermore, if the quadrature formula is the right Radau one (including 1 + n t ), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method . In addition , we present a generalization of DG that yield s a method identical to CG and collocation with arbitrary collocation points . Thus, the collocation , CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation . Finally, a ll scheme s discussed can be cast as s -stage implicit Runge -Kutta methods.