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In this paper, a compact and high order ADER (Arbitrary high order using DERivatives) scheme using the simple HWENO method (ADER-SHWENO) is proposed for hyperbolic conservation laws. The newly-developed method employs the Lax-Wendroff procedure to convert time derivatives to spatial derivatives, which provides the time evolution of the variables at the cell interfaces. This information is required for the simple HWENO reconstructions, which take advantages of the simple WENO and the classic HWENO. Compared with the original Runge-Kutta HWENO method (RK-HWENO), the new method has two advantages. Firstly, RK-HWENO method must solve the additional equations for reconstructions, which is avoided for the new method. Secondly, the SHWENO reconstruction is performed once with one stencil and is different from the classic HWENO methods, in which both the function and its derivative values are reconstructed with two different stencils, respectively. Thus the new method is more efficient than the RK-HWENO method. Moreover, the new method is more compact than the existing ADER-WENO method. Besides, the new method makes the best use of the information in the ADER method. Thus, the time evolution of the cell averages of the derivatives is simpler than that developed in the work [Li et. al., 447 (2021), 110661.]. Numerical tests indicate that the new method can achieve high order for smooth solutions both in space and time, keep non-oscillatory at discontinuities.

A high-order quasi-conservative discontinuous Galerkin (DG) method is proposed for the numerical simulation of compressible multi-component flows. A distinct feature of the method is a predictor-corrector strategy to define the grid velocity. A Lagrangian mesh is first computed based on the flow velocity and then used as an initial mesh in a moving mesh method (the moving mesh partial differential equation or MMPDE method ) to improve its quality. The fluid dynamic equations are discretized in the direct arbitrary Lagrangian-Eulerian framework using DG elements and the non-oscillatory kinetic flux while the species equation is discretized using a quasi-conservative DG scheme to avoid numerical oscillations near material interfaces. A selection of one- and two-dimensional examples are presented to verify the convergence order and the constant-pressure-velocity preservation property of the method. They also demonstrate that the incorporation of the Lagrangian meshing with the MMPDE moving mesh method works well to concentrate mesh points in regions of shocks and material interfaces., Comment: 44 pages, 71 figures

In this paper, a high order quasi-conservative discontinuous Galerkin (DG) method using the non-oscillatory kinetic flux is proposed for the 5-equation model of compressible multi-component flows with Mie-Gr\"uneisen equation of state. The method mainly consists of three steps: firstly, the DG method with the non-oscillatory kinetic flux is used to solve the conservative equations of the model; secondly, inspired by Abgrall's idea, we derive a DG scheme for the volume fraction equation which can avoid the unphysical oscillations near the material interfaces; finally, a multi-resolution WENO limiter and a maximum-principle-satisfying limiter are employed to ensure oscillation-free near the discontinuities, and preserve the physical bounds for the volume fraction, respectively. Numerical tests show that the method can achieve high order for smooth solutions and keep non-oscillatory at discontinuities. Moreover, the velocity and pressure are oscillation-free at the interface and the volume fraction can stay in the interval [0,1]., Comment: 41 pages, 70 figures

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 theoretically predicted order of convergence 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., Comment: 41 pages

A hybrid LDG-HWENO scheme is proposed for the numerical solution of KdV-type partial differential equations. It evolves the cell averages of the physical solution and its moments (a feature of Hermite WENO) while discretizes high order spatial derivatives using the local DG method. The new scheme has the advantages of both LDG and HWENO methods, including the ability to deal with high order spatial derivatives and the use of a small number of global unknown variables. The latter is independent of the order of the scheme and the spatial order of the underlying differential equations. One and two dimensional numerical examples are presented to show that the scheme can attain the same formal high order accuracy as the LDG method.