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Abstract: The local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection–diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters, which is termed as Runge–Kutta LDG (RKLDG) when TVD Runge–Kutta method is applied for time discretization. It has the advantage of flexibility in handling complicated geometry, h-p adaptivity, and efficiency of parallel implementation and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially non-oscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper, we investigated RKLDG methods with WENO and Hermite WENO (HWENO) limiters for solving convection–diffusion equations on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition. Numerical results are provided to illustrate the behavior of these procedures. [Copyright &y& Elsevier]

Abstract: A key idea in finite difference weighted essentially non-oscillatory (WENO) schemes is a combination of lower order fluxes to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is crucial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious oscillation. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. In this paper, we investigate hybrid schemes of WENO schemes with high order up-wind linear schemes using different discontinuity indicators and explore the possibility in avoiding the local characteristic decompositions and the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong shocks. The idea is to identify discontinuity by an discontinuity indicator, then reconstruct numerical flux by WENO approximation in discontinuous regions and up-wind linear approximation in smooth regions. These indicators are mainly based on the troubled-cell indicators for discontinuous Galerkin (DG) method which are listed in the paper by Qiu and Shu (J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially non-oscillatory limiters, SIAM Journal of Scientific Computing 27 (2005) 995–1013). The emphasis of the paper is on comparison of the performance of hybrid scheme using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance of hybrid scheme to save computational cost. Detail numerical studies in one- and two-dimensional cases are performed, addressing the issues of efficiency (less CPU time and more accurate numerical solution), non-oscillatory property. [ABSTRACT FROM AUTHOR]

• A new type of multi-resolution WENO limiter has been designed for Runge-Kutta discontinuous Galerkine (RKDG) schemes. • The limiter uses the information of the DG solution essentially only within the troubled cell itself, hence is very local. • Numerical examples are provided to demonstrate the good performance of this new limiter. In this paper, a new type of multi-resolution weighted essentially non-oscillatory (WENO) limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods is designed. This type of multi-resolution WENO limiters is an extension of the multi-resolution WENO finite volume and finite difference schemes developed in [43]. Such new limiters use information of the DG solution essentially only within the troubled cell itself, to build a sequence of hierarchical L 2 projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, fourth-order, and fifth-order RKDG methods with these multi-resolution WENO limiters have been developed as examples, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original DG methods. Benchmark examples are given to demonstrate the good performance of these RKDG methods with the associated multi-resolution WENO limiters. [ABSTRACT FROM AUTHOR]

An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation. [ABSTRACT FROM AUTHOR]

In this paper, we construct a high order weighted essentially non-oscillatory (WENO) finite difference discretization for compressible Navier-Stokes (NS) equations, which is rendered positivity-preserving of density and internal energy by a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The novelty of this paper is WENO reconstruction performed on variables from a positivity-preserving convection diffusion flux splitting, which is different from conventional WENO schemes solving compressible NS equations. The core advantages of our proposed method are robustness and efficiency, which especially are suitable for solving tough demanding problems of both compressible Euler and NS equation including low density and low pressure flow regime. Moreover, in terms of computational cost, it is more efficient and easier to implement and extend to multi-dimensional problems than the positivity-preserving high order discontinuous Galerkin schemes and finite volume WENO scheme for solving compressible NS equations on rectangle domain. Benchmark tests demonstrate that the proposed positivity-preserving WENO schemes are high order accuracy, efficient and robust without excessive artificial viscosity for demanding problems involving with low density, low pressure, and fine structure. • A positivity-preserving high order accurate finite difference WENO scheme for compressible N-S equations. • The method is the high order accuracy and positivity-preserving. • WENO reconstruction is performed on variables from a positivity-preserving convection diffusion flux splitting. • The method is robust and suitable for solving the compressible flow with low density and low pressure. [ABSTRACT FROM AUTHOR]

In this paper, a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving hyperbolic conservation laws on structured meshes. Here we only use the information defined on a hierarchy of nested central spatial stencils but do not introduce any equivalent multi-resolution representation, the terminology of multi-resolution HWENO follows that of the multi-resolution WENO schemes (Zhu and Shu, 2018) [29]. The main idea of our spatial reconstruction is derived from the original HWENO schemes (Qiu and Shu, 2004) [19] , in which both the function and its first-order derivative values are evolved in time and used in the reconstruction. Our HWENO schemes use the same large stencils as the classical HWENO schemes which are narrower than the stencils of the classical WENO schemes for the same order of accuracy. Only the function values need to be reconstructed by our HWENO schemes, the first-order derivative values are obtained from the high-order linear polynomials directly. Furthermore, the linear weights of such HWENO schemes can be any positive numbers as long as their sum equals one, and there is no need to do any modification or positivity-preserving flux limiting in our numerical experiments. Extensive benchmark examples are performed to illustrate the robustness and good performance of such finite volume and finite difference HWENO schemes. • High-order multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed. • Both finite volume and finite difference schemes are presented. • The new methods are robust, there are no need any additional limiter. [ABSTRACT FROM AUTHOR]

• A high order finite difference scheme for compressible two-medium flows. • The scheme is conservative. • The new method can maintain high order accuracy when the solution is smooth. • WENO is applied to the primitive variables to maintain the equilibrium of velocity and pressure across the interface. In this paper, a high order finite difference conservative scheme is proposed to solve two-medium flows. Our scheme has four advantages: First, our scheme is conservative, which is important to ensure the numerical solution captures the main features properly. Second, our scheme directly applies the WENO interpolation method to the primitive variables so that it can maintain the equilibrium of velocity and pressure across the interface, which is very helpful to obtain a non-oscillatory solution. Third, the usage of nodal values enables us to manipulate algebraic functions easily. Fourth, the scheme can maintain high order accuracy when the solution is smooth. Extensive numerical experiments are performed to verify the high resolution and non-oscillatory performance of this new scheme. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a high-order adaptive multi-resolution method on curvilinear grids for solving hyperbolic conservation laws. To leverage the success of high-order and high-resolution schemes on Cartesian grids, the governing equations and physical variables on curvilinear grids are transformed into Cartesian grids, resulting in the introduction of additional terms related to geometric metrics. In order to achieve high-order accuracy, two techniques are employed to eliminate errors caused by geometric metrics and preserve the property of Geometric Conservation Laws (GCLs). Firstly, a newly developed and simple technique is applied to remove metric-related errors in the dissipation part of the WENO-ZQ scheme on curvilinear grids. Secondly, a GCLs-preserving data transfer operator is utilized to avoid errors caused by the metrics in the adaptive multi-resolution (MR) method on curvilinear grids. By combining these two techniques, a newly developed GCLs-preserving high-order adaptive multi-resolution method on curvilinear grids is obtained. The high-order accuracy, high resolution, and efficiency of the developed method are demonstrated through several benchmark tests conducted in one and two dimensions. • A high-order adaptive multi-resolution method on curvilinear grids is proposed for the hyperbolic conservation laws. • The high-order geometric conservation laws preserving WENO-ZQ finite difference scheme on curvilinear grids are constructed. • The adaptive multi-resolution method on curvilinear grids is also designed to improve the resolution. • Numerical tests validate the good performance of the new method: high order and high resolution on curvilinear grids. [ABSTRACT FROM AUTHOR]