We describe a locally one-dimensional (LOD) time integration scheme for the diffusion equation in two space dimensions: ut= v(uxx+ uyy), based on the extended trapezoidal formula (ETF). The resulting LOD-ETF scheme is third order in time and is unconditionally stable. We describe the scheme for both Dirichlet and Neumann boundary conditions. We then extend the LOD-ETF scheme for nonlinear reaction-diffusion equations and for the convection-diffusion equation in two space dimensions. Numerical experiments are given to illustrate the obtained scheme and to compare its performance with the better-known LOD Crank-Nicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-ETF scheme provides both stable and accurate approximations for the true solution. © 2001 Elsevier Science Ltd.
Chawla, M. M., & Al-Zanaidi, M. A. (2001). An extended trapezoidal formula for the diffusion equation in two space dimensions. Computers and Mathematics with Applications, 42(1–2), 157–168. https://doi.org/10.1016/S0898-1221(01)00140-7