L1 scheme on graded mesh for subdiffusion equation with nonlocal diffusion term

Effect of Graded Mesh Number on the Solution of Convection-Diffusion Flow Problem with Quadratic Source

“…The efficacy of the approach is particularly demonstrated by the fact that the resultant difference scheme is stable and convergent, with convergence orders of 2 and 4 for time and space, respectively. The solution on uniform mesh, in contrary, gives lower order of convergence in maximum norm in time [18].…”

“…The usage of graded mesh is mainly important in finite element (FEM) [14][15][16][17][18], finite-difference (FDM) [19], exponential B-spline [14], and Newton methods [18]. In particular, the mesh is highly useful in the numerical experiment of reaction-diffusion problems [14,16,17], singularly perturbed problems with two parameters [15], sub-diffusion problems with nonlocal diffusion term [18], and evolution problems with a weakly singular kernel [19].…”

“…The usage of graded mesh is mainly important in finite element (FEM) [14][15][16][17][18], finite-difference (FDM) [19], exponential B-spline [14], and Newton methods [18]. In particular, the mesh is highly useful in the numerical experiment of reaction-diffusion problems [14,16,17], singularly perturbed problems with two parameters [15], sub-diffusion problems with nonlocal diffusion term [18], and evolution problems with a weakly singular kernel [19]. Non-graded mesh on which these problems are solved might include, for instance, a local algorithm which has been proposed [20] to obtain an optimal shape parameter for the infinitely smooth Radial Basis Functions (RBF) under grid-free environment.…”

“…In the second-order accurate numerical solution of partial differential evolution diffusion equations with nonlocal diffusion term, which generally exhibit a weakly singular kernel near the initial time, the use of L1 scheme on graded mesh was shown to be promising compared to uniform mesh [18,19]. A fully discrete difference scheme was constructed with space discretization by fourthorder accuracy compact difference method, while the Riemann-Liouville fractional integral was approximately calculated using the product integration approach for the time discretization, and a generalised second order accuracy Crank-Nicolson compact difference scheme for time-stepping was taken into consideration for smoothness of the solution at 𝑡 = 0.…”

“…First of all, it is regarded as a development of classic meshes like those in the Bakhvalov and Shishkin classes [14,16]. It has some desirable properties that the latter do not have [15], one of which is the achievement of optimum rate of convergence [18].…”