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The fractional Feynman-Kac equations describe the distribution of functionalsof non-Brownian motion, or anomalous diffusion, including two types called theforward and backward fractional Feynman-Kac equations, where the fractionalsubstantial derivative is involved. This paper focuses on the more widely usedbackward version. Based on the discretized schemes for fractional substantialderivatives proposed recently, we construct compact finite difference schemesfor the backward fractional Feynman-Kac equation, which has q-th (q=1, 2, 3, 4)order accuracy in temporal direction and fourth order accuracy in spatialdirection, respectively. In the case q=1, the numerical stability andconvergence of the difference scheme in the discrete L-infinity norm are provedstrictly, where a new inner product is defined for the theoretical analysis.Finally, numerical examples are provided to verify the effectiveness andaccuracy of the algorithms.

Maximum-norm error analysis of compact difference schemes for the backward fractional Feynman-Kac equation