An efficient adaptive mesh redistribution method for a non-linear Dirac equation

Abstract

This paper presents an efficient adaptive mesh redistribution method to solve a non-linear Dirac (NLD) equation. Our algorithm is formed by three parts: the NLD evolution, the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh. At each time level, the equidistribution principle is first employed to iteratively redistribute coarse mesh points, and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity. After an adaptive coarse mesh is generated ideally and finally, each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain, and then the solution vector is remapped on the resulting new fine mesh by an affine method. The NLD equation is finally solved by using a high resolution shock-capturing method on the (fixed) non-uniform fine mesh. Extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third-order rate of convergence, and yields an efficient and fast NLD solver that tracks and resolves both small, local and large solution gradients automatically.