Stabilized Linearly Implicit Simpson-type Schemes for Nonlinear Differential Equations

Abstract: The classical Simpson rule is an optimal fourth order two-step integration scheme for first-order initial-value problems; however, it is unconditionally unstable. An A-stabilized version of Simpson rule was given by Chawla et al [3] and an L-stable version was given by Chawla et al [2]. These rules are functionally implicit, and when applied for the time integration of nonlinear differential equations, require an iterative method such as Newton’s method for the solution of resulting nonlinear systems at each time step of integration. In the present paper, we present a new class of linearly implicit Simpson-type rules which are Aand L-stable. For the time integration of nonlinear differential equations, our linearly implicit schemes obviate the need to solve a nonlinear system at each time step of integration. The obtained schemes are computationally illustrated for stability and accuracy by considering a nonlinear initial value problem in ODEs and the diffusion equation with a nonlinear reaction term.