Accurate numerical resolution of transients in initial-boundary value problems for the heat equation
Flyer, N., & Fornberg, B. (2003). Accurate numerical resolution of transients in initial-boundary value problems for the heat equation. Journal Of Computational Physics, 184, 526-539. doi:10.1016/S0021-9991(02)00034-7
If the initial and boundary data for a PDE do not obey an infinite set of compatibility conditions, singularities will arise in the solution at the corners of the initial time–space domain. For dissipative equations, such as the 1-D heat equation or 1-D convection–diffusion equations, the impacts... Show moreIf the initial and boundary data for a PDE do not obey an infinite set of compatibility conditions, singularities will arise in the solution at the corners of the initial time–space domain. For dissipative equations, such as the 1-D heat equation or 1-D convection–diffusion equations, the impacts of these singularities are short lived. However, they can cause a very severe loss of numerical accuracy if we are interested in transient solutions. The phenomenon has been described earlier from a theoretical standpoint. Here, we illustrate it graphically and present a simple remedy which, with only little extra cost and effort, restores full numerical accuracy. Show less