Moving mesh generation for tracking a shock or steep moving front

For tracking a shock or steep moving front in the numerical solution of Partial Differential Algebraic Equations (PDAEs), an accurate spatial discretization method, Weighted Essentially Non-Oscillatory (WENO) scheme, is combined with moving grid techniques so that spacing of moving meshes is smoothed locally and globally. Several monitor functions, as metric criteria of node concentration, are examined. While the fixed grid method (uniform grid size) needs many mesh points to obtain enough solution accuracy, the moving grid method (non-uniform grid size) enhances accuracy even at small mesh numbers but it may be prohibitive owing to the addition of complex and non-linear mesh equations into physical PDAEs. The combination of the WENO scheme (based on an adaptive stencil idea) with the moving grid techniques improves stability and accuracy in the numerical solution over the commonly used moving grid method of central discretization. To locate adequate grid position in the moving mesh method, suitable monitor function according to problems must be selected.