Crank–Nicolson Method for the Advection-Diffusion EquationInvolving a Fractional Laplace Operator

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Résumé

We consider an advection-diffusion equation involving a fractional Laplace operator of order s 2 Š0; 1Š\f1=2g:. Using a combination of fractional left and right Riemann–Liouville derivatives of order 2s to approximate the fractional Laplace operator, we construct a numerical scheme using the Crank–Nicolson method. Using the Crank–Nicolson scheme, we succeeded in putting the numerical scheme of the problem under consideration in the form of a strictly and diagonally dominant positive definite matrix. This has allowed us to prove that the numerical scheme is stable and converges to first order in time and space for s 2 Š0; 1Š\f1=2g:.Numerical tests are performed to illustrate the results

Mots-clés

Crank–Nicolson method; fractional Laplace operator; fractional Riemann–Liouville derivatives; stability and convergence