Moving Mesh Method with Conservative Interpolation Based on L2-Projection

We develop an efficient one-dimensional moving mesh algorithm for solving partial differential equations.The main contribution of this paper is to design an effective interpolation scheme based on L2-projection for the moving mesh method.The proposed method preserves not only the mass-conservation but also the first order momentum of the underlying numerical solution at each mesh redistribution step.Numerical examples are presented to demonstrate the effectiveness of the new interpolation technique.

Numerical Simulation of Compressible Vortical Flows Using a ConservativeUnstructured-GridAdaptive Scheme

A two-dimensional numerical scheme for the compressible Euler equations is presented and applied here to the simulation of exemplary compressible vortical flows.The proposed approach allows to perform computations on unstructured moving grids with adaptation,which is required to capture complex features of the flowfield.Grid adaptation is driven by suitable error indicators based on theMach number and by element-quality constraints as well.At the new time level,the computational grid is obtained by a suitable combination of grid smoothing,edge-swapping,grid refinement and de-refinement.The grid modifications-including topology modification due to edge-swapping or the insertion/deletion of a new grid node-are interpreted at the flow solver level as continuous(in time)deformations of suitably-defined node-centered finite volumes.The solution over the new grid is obtained without explicitly resorting to interpolation techniques,since the definition of suitable interface velocities allows one to determine the new solution by simple integration of the Arbitrary Lagrangian-Eulerian formulation of the flow equations.Numerical simulations of the steady oblique-shock problem,of the steady transonic flow and of the start-up unsteady flow around the NACA 0012 airfoil are presented to assess the scheme capabilities to describe these flows accurately.

AnAdaptive Moving Mesh Method for Two-Dimensional Relativistic Hydrodynamics

This paper extends the adaptive moving mesh method developed by Tang and Tang[36]to two-dimensional(2D)relativistic hydrodynamic(RHD)equations.The algorithm consists of two“independent”parts:the time evolution of the RHD equations and the(static)mesh iteration redistribution.In the first part,the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations.The second part is an iterative procedure.In each iteration,the mesh points are first redistributed,and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way.Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.

Balanced Monitoring of Flow Phenomena in Moving Mesh Methods

Adaptive moving mesh research usually focuses either on analytical deriva-tions for prescribed solutions or on pragmatic solvers with challenging physical appli-cations. In the latter case, the monitor functions that steer mesh adaptation are oftendefined in an ad-hoc way. In this paper we generalize our previously used moni-tor function to a balanced sum of any number of monitor components. This avoidsthe trial-and-error parameter fine-tuning that is often used in monitor functions. Thekey reason for the new balancing method is that the ratio between the maximum andaverage value of a monitor component should ideally be equal for all components.Vorticity as a monitor component is a good motivating example for this. Entropy alsoturns out to be a very informative monitor component. We incorporate the monitorfunction in an adaptive moving mesh higher-order finite volume solver with HLLCfluxes, which is suitable for nonlinear hyperbolic systems of conservation laws. Whenapplied to compressible gas flow it produces very sharp results for shocks and otherdiscontinuities. Moreover, it captures small instabilities (Richtmyer-Meshkov, Kelvin-Helmholtz). Thus showing the rich nature of the example problems and the effective-ness of the new monitor balancing.