A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

• Published in: Engineering analysis with boundary elements 2023-12, Vol.157, p.413-423
• Main Authors: Soopramanien, Shilpa Selinska Gina, Thakoor, Nawdha, Tangman, Desiré Yannick, Bhuruth, Muddun
• Format: Article
• Language: English
• Subjects: Fractional diffusion; Higher-order Caputo approximations; Local radial point interpolation method; Weak-form algorithm
• ISSN: 0955-7997; 1873-197X
• DOI: 10.1016/j.enganabound.2023.09.021

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity. •New analytical formulas for two-dimensional LRPIM shape functions are derived.•Closed-form entries for LRPIM discretisation matrices are derived.•Graded meshes are employed for problems which have a layer at time zero.•Simplified L1-2 Caputo derivative formulas on graded meshes are derived.•Convergence of a strong-form algorithm on uniform meshes is established.