Two three-parallel and three-processor SDIRK methods for stiff initial-value problems

In this paper we propose two parallel diagonal iteration processes for solving two three-stage implict Runge-Kutta methods with stage order equal to 3. The resulting schemes can be regarded as parallel singly diagonally implicit Runge-Kutta methods (PSDIRK methods) which are A-stable and L-stable, respectively, when the classical linear test equation is used. When these methods are implemented on a three-processor computer, they require effectively the solution of three implicit relations (sequential time units) per step and per processor. The numerical experiments show the efficiency of these methods when they are compared with other sequential and parallel methods from the literature for solving some linear and nonlinear stiff initial-value problems.