A moving mesh method is proposed for solving reaction-diffusion equations. The finite element method is used to solving the partial different equation system, and an efficient numerical scheme is applied to implement mesh moving. In the practical calculations, the moving mesh step and the problem equation solver are performed alternatively. Several numerical examples are presented, including the Gray-Scott, the Activator-Inhibitor and a case with a growing domain. It is illustrated numerically that the moving mesh methods costs much lower, compared with the numerical schemes on a fixed mesh. Even in the case of complex pattern dynamics described by the reaction-diffusion systems, the adapted meshes can capture the details successfully.

AMG preconditioners are typically designed for partial differential equation solvers and divergence-interpolation in a moving mesh strategy. Here we introduce an AMG preconditioner to solve the unsteady Navier-Stokes equations by a moving mesh finite element method. A 4P1 – P1 element pair is selected based on the data structure of the hierarchy geometry tree and two-layer nested meshes in the velocity and pressure. Numerical experiments show the efficiency of our approach.

In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.

An efficient finite difference framework based on moving meshes methods is developed for the three-dimensional free surface viscoelastic flows. The basic model equations are based on the incompressible Navier-Stokes equations and the Oldroyd-B constitutive model for viscoelastic flows is adopted. A logical domain semi-Lagrangian scheme is designed for moving-mesh solution interpolation and convection. Numerical results show that harmonic map based moving mesh methods can achieve better accuracy for viscoelastic flows with free boundaries while using much less memory and computational time compared to the uniform mesh simulations.

We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L2-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method.