Convergence of a Fully Conservative Volume Corrected Characteristic Method for Transport Problems

We consider the convergence of a volume corrected characteristics-mixed method (VCCMM) for advection-diffusion systems. It is known that, without volume correction, the method is first order convergent, provided there is a nondegenerate diffusion term. We consider the advective part of the system and give some properties of the weak solution. With these properties we prove that the volume corrected method, with no diffusion term, gives a lower order L1-convergence rate of O(h/√Δt + h + (Δt)r), where r is related to the accuracy of the characteristic tracing. This result compares favorably to Godunov’s method, but avoids the CFL constraint, so large time steps can be taken in practice. In fact, Godunov’s method converges at O(h1/2), which is our result for Δt = Ch, where now C is not limited. However, the optimal choice, Δt = Ch2/(2r+1), gives a better rate, O(h2r/(2r+1)), than Godunov’s method, e.g., O(h2/3) if r = 1. With a nondegenerate diffusion term, we obtain an L2-error estimate for the problem. We also prove the existence of, and give an error estimate for, a perturbed velocity field for which the volume is locally conserved. Finally, some convergence tests are given to verify the optimal convergence rate for r = 1.