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In this paper, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme.

A long-standing and formidable challenge faced by all conservative schemes for relativistic magnetohydrodynamics (RMHD) is the recovery of primitive variables from conservative ones. This process involves solving highly nonlinear equations subject to physical constraints. An ideal solver should be "robust, accurate, and fast -- it is at the heart of all conservative RMHD schemes," as emphasized in [S.C. Noble et al., ApJ, 641:626-637, 2006]. Despite over three decades of research, seeking efficient solvers that can provably guarantee stability and convergence remains an open problem. This paper presents the first theoretical analysis for designing a robust, physical-constraint-preserving (PCP), and provably (quadratically) convergent Newton-Raphson (NR) method for primitive variable recovery in RMHD. Our key innovation is a unified approach for the initial guess, devised based on sophisticated analysis. It ensures that the NR iteration consistently converges and adheres to physical constraints. Given the extreme nonlinearity and complexity of the iterative function, the theoretical analysis is highly nontrivial and technical. We discover a pivotal inequality for delineating the convexity and concavity of the iterative function and establish theories to guarantee the PCP property and convergence. We also develop theories to determine a computable initial guess within a theoretical "safe" interval. Intriguingly, we find that the unique positive root of a cubic polynomial always falls within this interval. Our PCP NR method is versatile and can be seamlessly integrated into any RMHD scheme that requires the recovery of primitive variables, potentially leading to a broad impact in this field. As an application, we incorporate it into a discontinuous Galerkin method, resulting in fully PCP schemes. Several numerical experiments demonstrate the efficiency and robustness of the PCP NR method., Comment: 26 pages, 7 figures

In this paper, a fifth-order moment-based Hermite weighted essentially non-oscillatory scheme with unified stencils (termed as HWENO-U) is proposed for hyperbolic conservation laws. The main idea of the HWENO-U scheme is to modify the first-order moment by a HWENO limiter only in the time discretizations using the same information of spatial reconstructions, in which the limiter not only overcomes spurious oscillations well, but also ensures the stability of the fully-discrete scheme. For the HWENO reconstructions, a new scale-invariant nonlinear weight is designed by incorporating only the integral average values of the solution, which keeps all properties of the original one while is more robust for simulating challenging problems with sharp scale variations. Compared with previous HWENO schemes, the advantages of the HWENO-U scheme are: (1) a simpler implemented process involving only a single HWENO reconstruction applied throughout the entire procedures without any modifications for the governing equations; (2) increased efficiency by utilizing the same candidate stencils, reconstructed polynomials, and linear and nonlinear weights in both the HWENO limiter and spatial reconstructions; (3) reduced problem-specific dependencies and improved rationality, as the nonlinear weights are identical for the function $u$ and its non-zero multiple $\zeta u$. Besides, the proposed scheme retains the advantages of previous HWENO schemes, including compact reconstructed stencils and the utilization of artificial linear weights. Extensive benchmarks are carried out to validate the accuracy, efficiency, resolution, and robustness of the proposed scheme., Comment: 44 pages, 14 figures, 6 tables

The relativistic hydrodynamics (RHD) equations have three crucial intrinsic physical constraints on the primitive variables: positivity of pressure and density, and subluminal fluid velocity. However, numerical simulations can violate these constraints, leading to nonphysical results or even simulation failure. Designing genuinely physical-constraint-preserving (PCP) schemes is very difficult, as the primitive variables cannot be explicitly reformulated using conservative variables due to relativistic effects. In this paper, we propose three efficient Newton--Raphson (NR) methods for robustly recovering primitive variables from conservative variables. Importantly, we rigorously prove that these NR methods are always convergent and PCP, meaning they preserve the physical constraints throughout the NR iterations. The discovery of these robust NR methods and their PCP convergence analyses are highly nontrivial and technical. As an application, we apply the proposed NR methods to design PCP finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the RHD equations. Our PCP HWENO schemes incorporate high-order HWENO reconstruction, a PCP limiter, and strong-stability-preserving time discretization. We rigorously prove the PCP property of the fully discrete schemes using convex decomposition techniques. Moreover, we suggest the characteristic decomposition with rescaled eigenvectors and scale-invariant nonlinear weights to enhance the performance of the HWENO schemes in simulating large-scale RHD problems. Several demanding numerical tests are conducted to demonstrate the robustness, accuracy, and high resolution of the proposed PCP HWENO schemes and to validate the efficiency of our NR methods., Comment: 49 pages

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.

We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass conservation naturally holds due to the weak formulation of the characteristic Galerkin method and the design of the HWENO reconstructions. We apply a positive-preserving limiter to maintain the positivity of numerical solutions when needed. Although the proposed SL framework allows us to take large time steps for improving computational efficiency, it also brings challenges to the spatial reconstruction technique; we construct two kind of novel HWENO reconstructions to fit the need for the proposed SL framework. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model -- a generalization of the shallow water equations that accounts for effects of water temperature variations. Thermodynamic processes are important particularly in the upper layers of the ocean where the variations of sea surface temperature play a fundamental role in climate change. The well-balance property which requires numerical schemes to preserve the lake-at-rest steady state is crucial to the simulation of perturbation waves over that steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance, positivity-preserving, and high-order properties, a DG-interpolation scheme (with or without scaling positivity-preserving limiter) and special treatments pertaining to the Ripa model are employed in the transfer of both the flow variables and bottom topography from the old mesh to the new one and in the TVB limiting process. Mesh adaptivity is realized using an MMPDE moving mesh approach and a metric tensor based on an equilibrium variable and water depth. A motivation is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom structure). Numerical examples in one and two dimensions are presented to demonstrate the well-balance, high-order accuracy, and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state., Comment: 37 pages, 24 figures. arXiv admin note: substantial text overlap with arXiv:2006.15187

In this paper, a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J. Comput. Phys., 446 (2021) 110653], in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries. However, in this paper, only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments. In addition, an extra modification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolution near discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights can also be any positive numbers as long as their sum equals one and the CFL number can still be 0.6 whether for the one or two dimensional case. Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme., Comment: 36pages, 8 figures