An extended trapezoidal formula for the diffusion equation

A finite-difference scheme for the diffusion equation that has enjoyed great popularity is the Crank-Nicolson scheme [1] based on the classical trapezoidal formula for integration in time. For problems with discontinuities in the boundary conditions and the initial conditions, the Crank-Nicolson scheme can give unwanted oscillations in the computed solution. We present an alternative scheme based on the extended trapezoidal formula for integration in time. The resulting Extended Trapezoidal Formula Finite Difference Scheme (ETF-FDS) is third order in time, unconditionally stable and, unlike the Crank-Nicolson scheme, ETF-FDS can cope with discontinuities in the boundary conditions and the initial conditions as demonstrated by the numerical examples considered.