Optimal and near-optimal advection-diffusion finite-difference schemes. II. Unsteadiness and non-uniform grid

It is shown how time-dependent coefficients and non-uniform unsteady grid spacing modifies the optimal weights in a 3 spatial-points, 2 time-step, implicit approximation for the numerical solution of the equation λtc+λc+uλxck2xc=q+λxf+2xg. At the grid points the extremely high order of approximation for the numerical solutions is such that if loss of accuracy is to be avoided, then interpolation must use values extending beyond the local 3-by-2 computational module. Illustrative examples show that reasonable accuracy is possible with extremely long time-steps on sparse non-uniform moving grids.