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The local discontinuous Galerkin method for convection-diffusion-fractional anti-diffusion equations. (English) Zbl 1447.65070
Summary: In this paper, we consider the discontinuous Galerkin method for solving time dependent partial differential equations with convection-diffusion terms and anti-diffusive fractional operator of order $$\alpha \in(1, 2)$$. These equations are motivated by two distinct applications: a dune morphodynamics model and a signal filtering model. The key to study these numerical schemes is to split the anti-diffusive operators into a singular and non-singular integral representations. The problem is then expressed as a system of low order differential equations and a local discontinuous Galerkin method is proposed for these equations. We prove nonlinear stability estimates and optimal order of convergence $$\mathcal{O}({\Delta} x^{k + 1})$$ for linear equations and an order of convergence of $$\mathcal{O}({\Delta} x^{k + \frac{1}{2}})$$ for the nonlinear problem. Finally numerical experiments are given to illustrate qualitative behaviors of solutions for both applications and to confirme our convergence results.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals
FODE
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