Variable step-size implicit-explicit linear multistep methods for time-dependent PDEs

The objectives of this thesis are to design, analyze and numerically investigate easily implementable Variable Step-Size Implicit-Explicit (VSIMEX) Linear Multistep Methods for time-dependent PDEs. The thesis begins with a derivation of the family of second-order, two-step VSIMEX schemes with two free parameters. A zero-stability analysis of these VSIMEX schemes gives analytical results on the restriction of the step-size ratio for general second-order VSIMEX schemes. The family of third-order, three-step VSIMEX schemes with three free parameters is also derived. A zero-stability analysis of these VSIMEX schemes gives numerical values for the step-size restrictions. A fourth-order, four-step VSIMEX scheme and its stability properties are also studied. Numerically, we apply our new VSIMEX schemes to the 1-D advection-diffusion and Burgers' equations. The expected orders of convergence are achieved, and accurate approximate solutions are obtained. Our results demonstrate the superiority of VSIMEX schemes over classical IMEX schemes in solving Burgers' equation.