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Stability and convergence of 3-point WSGD schemes for two-sided space fractional advection-diffusion equations with variable coefficients. (English) Zbl 1481.65144

Summary: In this paper, we consider high order numerical methods for the solution of the initial-boundary value problem of two-sided space fractional advection-diffusion equations (SFADEs). We use the Crank-Nicolson (CN) technique to discretize the temporal derivative, and apply 3-point weighted and shifted Grünwald difference (WSGD) operators to discretize the fractional derivatives of order \(\alpha \in(1, 2)\) and fractional derivatives of order \(\beta \in(0, 1)\), respectively. As a result, a new family of CN-WSGD schemes for SFADEs with temporally 2nd order and spatially \(j\)th order (\(j \geq 2\)) accuracy are obtained. We then analyse the stability and the convergence of the numerical schemes. The extrapolated WSGD (EWSGD) operators are also discussed. Numerical examples are implemented to verify the theoretical results.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65B05 Extrapolation to the limit, deferred corrections
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
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