Adaptively Generated Moving Mesh Algorithms for Singularly Perturbed Parabolic Problems with a Large Lag in Time

A Robust ADI-Based Direct Discontinuous Galerkin Method for Two-Dimensional Singularly Perturbed Parabolic PDEs

This study proposes two-dimensional one-parameter singularly perturbed parabolic problems and develops an operator-splitting direct discontinuous Galerkin (DDG) method for their numerical solution. To improve computational efficiency, an alternating direction implicit (ADI) splitting approach decomposes the original tw...

Reduction of a cell centered finite volume scheme

Adaptive moving mesh methods for isotropic/anisotropic mean curvature flow with axisymmetric geometry

This paper introduces adaptive moving mesh methods for the numerical simulation of axisymmetric mean curvature flow, addressing both isotropic and anisotropic cases. The methods are developed within the framework of the mesh equidistribution principle, where a carefully designed tangential velocity is employed to dynam...

Some new numerical schemes for delay sub-diffusion equations via the idea of symmetry

In this work, we propose several novel numerical schemes that synergistically integrates the $L1$ method and symmetric fractional-order reduction (SFOR) method, specifically designed for fractional sub-diffusion equations with time delay. Crucially, we rigorously establish an optimal convergence order of $2 - \alpha/2$...

A high-order interpolation method for hanging points based on the adaptive Cartesian mesh

The adaptive mesh refinement technology is widely employed in Cartesian grid methods to improve computational efficiency by refining the mesh only in regions requiring high resolution. However, local refinement introduces hanging points at coarse–fine interfaces, making the construction of accurate and robust interpola...

An energy stable and accuracy-preserving finite volume scheme based on the SAV method with application to wall-distance computation

A novel semi-implicit second-order finite volume scheme integrating the scalar auxiliary variable (SAV) approach is proposed for solving the pseudo-time Eikonal equation in wall-distance computation. Unconditional energy stability under zero boundary conditions is rigorously proved, eliminating the dependence of the ti...