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• A moving mesh discontinuous Galerkin method is presented for conservation laws. • The method is a combination of the DG method and the moving mesh PDE. • No interpolation is needed for physical variables from the old mesh to the new one. • The method achieves the theoretically predicted order of convergence for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic conservation laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system conservation laws are presented. It is shown that the moving mesh DG method achieves the second and third order of convergence for P 1 and P 2 elements, respectively, for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated. [ABSTRACT FROM AUTHOR]

A new functional is presented for variational mesh generation and adaptation. It is formulated based on combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter. The functional is shown to be coercive but not convex. A solution procedure using a discrete moving mesh partial differential equation is employed. It is shown that the element volumes and altitudes of a mesh trajectory of the mesh equation associated with the new functional are bounded away from zero and the mesh trajectory stays nonsingular if it is so initially. Numerical examples demonstrate that the new functional performs comparably as an existing one that is also based on the equidistribution and alignment conditions and known to work well but contains an additional parameter. [ABSTRACT FROM AUTHOR]

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the decomposition of the strain tensor into tensile and compressive components is essential for the success of the phase-field modeling of brittle fracture but results in a non-smooth elastic energy and stronger nonlinearity in the governing equation. This makes the governing equation much more difficult to solve and, in particular, Newton's iteration often fails to converge. Three regularization methods are proposed to smooth out the decomposition of the strain tensor. Numerical examples of fracture propagation under quasi-static load demonstrate that all of the methods can effectively improve the convergence of Newton's iteration for relatively small values of the regularization parameter but without compromising the accuracy of the numerical solution. They also show that the moving mesh finite element method is able to adaptively concentrate the mesh elements around propagating cracks and handle multiple and complex crack systems. [ABSTRACT FROM AUTHOR]

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a $$C^0$$ finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution. [ABSTRACT FROM AUTHOR]

We present a simple direct discretization for functionals used in the variational mesh generation and adaptation. Meshing functionals are discretized on simplicial meshes and the Jacobian matrix of the continuous coordinate transformation is approximated by the Jacobian matrices of affine mappings between elements. The advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional, which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional. Moreover, the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form, which allows a simple implementation of variational mesh generation and adaptation on computer. Since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods, the result in this work can be used to develop simple implementations of those methods. Numerical examples are given. [ABSTRACT FROM AUTHOR]

This paper is concerned with moving mesh finite difference solution of partial differential equations. It is known that mesh movement introduces an extra convection term and its numerical treatment has a significant impact on the stability of numerical schemes. Moreover, many implicit second and higher order schemes, such as the Crank–Nicolson scheme, will lose their unconditional stability. A strategy is presented for developing temporally high-order, unconditionally stable finite difference schemes for solving linear convection–diffusion equations using moving meshes. Numerical results are given to demonstrate the theoretical findings. [ABSTRACT FROM PUBLISHER]

Abstract: The stability of three moving-mesh finite-difference schemes is studied in the norm for one-dimensional linear convection–diffusion equations. These schemes use central finite differences for spatial discretization and the method for temporal discretization, and they are based on conservative and non-conservative forms of transformed partial differential equations. The stability conditions obtained consist of the CFL condition and the mesh speed related conditions. The CFL condition is independent of the mesh speed and has the same form as that for fixed meshes. The mesh speed related conditions restrict how fast the mesh can move. The conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine. Illustrative numerical results are presented. [Copyright &y& Elsevier]

Abstract: A new concept called the dominance of equidistribution is introduced for analyzing moving mesh partial differential equations for numerical simulation of blowup in reaction diffusion equations. Theoretical and numerical results show that a moving mesh method works successfully when the employed moving mesh equation has the dominance of equidistribution. The property can be verified using dimensional analysis. In several aspects the current work generalizes previous work where a moving mesh equation is shown to have this dominance of equidistribution if it preserves the scaling invariance of the underlying physical partial differential equation and uses a small, constant value for (a parameter used for adjusting response time of the mesh movement to the change in the physical solution). Also, cases with both constant and variable are considered here. [Copyright &y& Elsevier]