Extended one-step time-integration schemes for convection-diffusion equations

We first describe a one-parameter family of unconditionally stable third-order time-integration schemes for the convection-diffusion equation: ut + cux = vuxx, based on the extended trapezoidal formulas of Usmani and Agarwal [1]. Interestingly, there exists a method that is fourth order as well as unconditionally stable. We then describe a one-parameter family of unconditionally stable fourth-order time-integration schemes, based on the extended Simpson rules of Chawla et al. [2]. Again, there exists a method that is fifth order as well as unconditionally stable. The stability and accuracy of the obtained methods are tested, and compared with the widely used method of Crank-Nicolson, by considering three problems of practical interest.