We propose an explicit, single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian–Eulerian approach for one-dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order $ \theta $ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional diffusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with $O(h^{k+1}+\varDelta t^2)$ is derived, where $k\geqslant 0$ denotes the index of the basis function. Extensive numerical results with $Q^k(k=0,1,2,3)$ elements are provided to confirm our theoretical results, which also show that the second-order convergence rate in time is not impacted by the changed parameter $\theta$.

The ${\mathcal {H}}$-tensor is a new developed concept in tensor analysis and it is an extension of the ${\mathcal {M}}$-tensor. In this paper, we present some criteria for identifying nonsingular ${\mathcal {H}}$-tensors and give two numerical examples.

The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equation. The time variable has been discretized by a second-order finite difference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confirm the theoretical results.

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving variable-coefficient one-dimensional (steady state) fractional differential equations (DEs) with two-sided fractional derivatives (FDs). The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided FD when the right-sided FD is approximated by two consecutive applications of the first-order backward Euler method. Our scheme reduces to the standard second-order central difference in the absence of FDs. The existence and uniqueness of the numerical solution are proved, and truncation errors of order h are demonstrated (h denotes the maximum space step size). The numerical tests illustrate the global O(h) accuracy, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

For two-dimensional (2D) time fractional diffusion equations, we construct a numerical method based on a local discontinuous Galerkin (LDG) method in space and a finite difference scheme in time. We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable. Numerical results indicate the effectiveness and accuracy of the method and confirm the analysis.