Maximum modulus principle estimates for one dimensional fractional diffusion equation

Abstract

We present scheme I for solving one-dimensional fractional diffusion equation with variable coefficients based on the maximum modulus principle and two Grunwald approximations. Scheme II is obtained by using classic Crank-Nicolson approximations in order to improve the time convergence. Schemes are proved to be unconditionally stable and second-order accuracy in spatial grid size for the problem with order of fractional derivative belonging to $$\left[ {\left( {\sqrt {17} - 1} \right)/2,2} \right]$$ using the maximum modulus principle. A numerical example is given to confirm the theoretical analysis result.