Generalized trapezoidal formulas for convection-diffusion equations

Abstract

We describe new time-integration schemes for the linear convection-diffusion equation and for the (viscous) Burgers' equation, employing the generalized trapezoidal formulas of Chawla et al [2]. The obtained generalized trapezoidal formula finite-difference schemes (GTF(α)-FDS) are second order in both time and space and unconditionally stable. The better known existing schemes employ the Euler, the backward Euler or the classical arithmetic-mean trapezoidal formula (AM-TF) for integration in time. The performance of our present GTF(α)-FDS is compared with the AM-TF scheme by considering three test problems wherein the significance of the role played by the parameter α becomes evident in providing both stability and accuracy of the computed solution in the presence of diffusivity.