Extended one-step time-integration schemes for convection-diffusion equations (Q1570125)

Linear spatially one-dimensional convection-diffusion equation of the type \(u_t +c u_x = \nu u_{xx}\) is considered in this paper especially from the numerical point of view. A one-parameter family of unconditionally stable third-order time integration schemes based on extended trapezoidal formulas of Usmani and Agarwal is described. Unconditionally stable fourth order time integration schemes based on extended Simpson rules of Chawla, Karaballi and Al-Sahhar are also investigated together with a method that is fifth order as well as unconditionally stable. Problems of practical interest are also included. The presented methods are there tested and compared with the well-known Crank-Nicolson method, esspecially their stability and accuracy.