LDG approximation of a nonlinear fractional convection-diffusion equation using B-spline basis functions. (English) Zbl 1518.65110
Summary: This paper develops new numerical schemes for solution to nonlinear fractional convection-diffusion equations of order \(\beta\in[1,2]\). We propose the local discontinuous Galerkin methods by adopting linear, quadratic, and cubic B-spline basis functions and prove stability and optimal order of convergence \(O(h^{k+1})\) for the fractional diffusion problem. This method transforms the equation into a system of first-order equations and approximates the solution of the equation by selecting the appropriate basis functions. The B-Spline functions significantly improve the accuracy and stability of the method. The performed numerical results demonstrate the efficiency and accuracy of the proposed scheme in different conditions and confirm the optimal order of convergence.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65D07 | Numerical computation using splines |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
local discontinuous Galerkin method; convection-diffusion equation; fractional derivatives; B-spline basis functions; stabilitySoftware:
SplinePakReferences:
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