Abstract
In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.
Highlights
To test the robustness of the numerical method and to assess the theoretical convergence orders derived in the previous section, we considered three different β values of
We showed that finite difference schemes on uniform grids provide results with the loss of the convergence order
We proposed an alternative numerical method based on time nonuniform meshes
Summary
In order to numerically solve the problem (1), (6), (7), we start with the discretization of the integral in (1). In order to approximate the spatial derivative, we consider a uniform mesh, on the interval. Substituting (14) and (15) in (1) and denoting Ui (t) ≈ u(t, xi ), we obtain the semidiscretized scheme:. Substituting in (16) and denoting by Uik ≈ u(tk , xi ), we obtain the finite difference scheme:. In order to obtain a numerical solution for (1), (6), and (7), we need to solve the linear systems of Equation (21) noting that, taking (17) and (18) into account, we have: U0l = u0 , ULl = u L , Ui0.
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