Multigrid methods for time fractional conservation laws

Abstract

For solving the time fractional conservation laws with discontinuous solutions such as shock waves, current feasible methods are the implicit finite-volume TVD schemes which employ the Lax-Friedrichs fluxes corrected by the limited slopes (see e.g. [12]). However, the schemes are hard to implement when the fractional order α is close to zero since in their implementation there are still no efficient methods to solve the strongly nonlinear spacial discrete systems at discrete times. We, aiming at increasing the implementation efficiency of the schemes above, develop two multigrid methods to solve these nonlinear spacial discrete systems. Numerical tests show that both multigrid methods have good convergence for relatively larger α, and the second one has an advantage that its convergence is affected slightly when the order α decreases to zero.