Hermite WENO Research Articles

High-order oscillation-eliminating Hermite WENO method for hyperbolic conservation laws

This paper proposes high-order accurate, oscillation-eliminating, Hermite weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for hyperbolic conservation laws, motivated by the oscillation-eliminating (OE) discontinuous Galerkin schemes recently proposed in [M. Peng, Z. Sun, and K. Wu, 2024 [30]]. The OE-HWENO schemes incorporate an OE procedure after each Runge–Kutta stage, by dampening the first-order moments of the HWENO solution to suppress spurious oscillations without any problem-dependent parameter. The OE procedure acts as a moment filter and is derived from the solution operator of a novel damping equation, which is exactly solved without any discretization. Thanks to this distinctive feature, the OE-HWENO method remains stable with a normal CFL number, even for strong shocks resulting in highly stiff damping terms. To ensure the essentially non-oscillatory property of the OE-HWENO method across problems with varying scales and wave speeds, we design a scale-invariant and evolution-invariant damping equation and propose a generic dimensionless transformation for HWENO reconstruction. The OE-HWENO method offers several notable advantages over existing HWENO methods. First, the OE procedure is highly efficient and straightforward to implement, requiring only simple multiplication of first-order moments by a damping factor. Furthermore, we rigorously prove that the OE procedure maintains the high-order accuracy and local compactness of the original HWENO schemes and demonstrate that it does not compromise the spectral properties via the approximate dispersion relation for smooth solutions. Notably, the proposed OE procedure is non-intrusive, enabling seamless integration as an independent module into existing HWENO codes. Finally, we rigorously analyze the bound-preserving (BP) property of the OE-HWENO method using the optimal cell average decomposition approach [S. Cui, S. Ding, and K. Wu, 2024 [8]], which relaxes the theoretical BP constraint for time step-size and reduces the number of decomposition points, thereby further enhancing efficiency. Extensive benchmarks validate the accuracy, efficiency, high resolution, and robustness of the OE-HWENO method.

Fourth-Order Conservative Non-splitting Semi-Lagrangian Hermite WENO Schemes for Kinetic and Fluid Simulations

AbstractWe 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. The new HWENO reconstructions are meticulously designed to strike a delicate balance between curbing numerical oscillation and introducing excessive dissipation. Mass conservation naturally holds due to the weak formulation of the semi-Lagrangian discontinuous Galerkin method and the design of the HWENO reconstructions. We apply a positivity-preserving limiter to maintain the positivity of numerical solutions when needed. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes I: Steady flows

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows.

Compact high-order gas-kinetic scheme for direct numerical simulation of compressible turbulent flows

In this paper, the three-dimensional fully compact high-order gas-kinetic scheme (HGKS) is proposed for the direct numerical simulation of compressible turbulent flows. Because of the high-order gas evolution model, the numerical fluxes as well as the point-wise conservative variables can be evaluated from the time-accurate gas distribution function at the cell interface. As a result, both the cell-averaged variables and their cell-averaged gradients can be updated inside each cell. Based on the cell averaged values and their gradients, the compact Hermite weighted essentially non-oscillatory (HWENO) scheme is developed, in which the dimension-by-dimension reconstruction is used for three-dimensional turbulences. In both normal and tangential directions, the fifth-order HWENO reconstruction is adopted. Compared with the classical WENO scheme, the stencil for the HWENO scheme only contains 33 cells for each cell. To achieve the temporal accuracy, the two-stage fourth-order temporal discretization is used. For the evaluation of point-wise variables, the simplified third-order gas-kinetic solver is used. Several classical benchmark problems are simulated, which validate the accuracy, resolution, and robustness of compact HGKS. As a comparison, the numerical results of HGKS using non-compact WENO reconstruction are also provided. Due to the compact stencil, the compact HGKS has a favorable performance for turbulence simulation in resolving the multi-scale structures.

Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics

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. Our NR methods are versatile and can be seamlessly incorporated into any RHD schemes that require the recovery of primitive variables. As an application, we apply them 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.

Finite-Difference Hermite WENO Scheme for Degasperis-Procesi Equation

We present a fifth-order finite-difference Hermite weighted essentially non-oscillatory (HWENO) method for solving the Degasperis–Procesi (DP) equation in this paper. First, the DP equation can be rewritten as a system of equations consisting of hyperbolic equations and elliptic equations by introducing an auxiliary variable, since the equations contain nonlinear higher order derivative terms. Then, the auxiliary variable equations are solved using the Hermite interpolation, while the HWENO scheme is performed for the hyperbolic equations. Compared with the popular WENO-type scheme, the most important feature of the HWENO scheme mentioned in this paper is the compactness of its spatial reconstruction stencil, which can achieve the fifth-order accuracy of the expected design with only three points, while the WENO method requires five points. Finally, we demonstrate the effectiveness of the HWENO method in various aspects by conducting some benchmark numerical tests.

A robust fifth order finite difference Hermite WENO scheme for compressible Euler equations

In this paper, we develop a robust fifth order finite difference Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations following the HWENO with limiter (HWENO-L) scheme (Zhang and Zhao, 2023). The HWENO-L scheme reduced storage and increased efficiency by using restricted derivatives only for time discretizations. However, it cannot control spurious oscillations well when facing strong shocks since the derivatives are directly used in spatial discretizations without any restrictions. To address such an issue, our proposed HWENO scheme performs flux reconstructions in the finite difference framework without using the derivative value of a target cell, which can result in a simpler and more robust scheme. The resulting scheme is simpler while achieving fifth order accuracy, making it more efficient. Besides, numerically we find it is very robust for some extreme problems even without positivity-preserving limiters. The proposed scheme also inherits advantages of previous HWENO schemes, including arbitrary positive linear weights in flux reconstructions, compact reconstructed stencils, and high resolution. Extensive numerical tests are performed to demonstrate the fifth order accuracy, efficiency, robustness, and high resolution of the proposed HWENO scheme.

High Order Finite Difference Hermite WENO Fast Sweeping Methods for Static Hamilton-Jacobi Equations

High Order Finite Difference Hermite WENO Fast Sweeping Methods for Static Hamilton-Jacobi Equations

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balanced property, we adopt the similar idea of WENO-XS scheme (Xing and Shu (2005) [30]) to balance the flux gradients and the source terms. The fluxes in the original equations are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balanced property. Instead of using a five-point stencil in the fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction. Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.

New finite difference unequal-sized Hermite WENO scheme for Navier-Stokes equations

In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spatial reconstruction procedure uses a convex combination of one quintic polynomial and two linear polynomials defined on three unequal-sized central or biased spatial stencils. Compared with the classical finite difference, finite volume, or mixed finite difference and finite volume HWENO schemes [32] , [33] , the new scheme uses same largest stencil and only three stencils in spatial reconstruction procedures, and the first-order derivative values in auxiliary equations can be obtained directly rather than using complex HWENO-type reconstruction methodologies [32] , [33] . Moreover, the associated viscosity terms could be directly computed from the function values and first-order derivative values by using the characteristic of the WENO-type spatial reconstructions. Meanwhile, the linear weights can be any positive numbers on condition that their summation equals one, which avoids the problem of negative linear weights in classical HWENO schemes [24] , [25] . So the new finite difference US-HWENO scheme is more suitable for solving Navier-Stokes equations, and can be easily implemented to multi-dimensions on structured meshes. Some benchmark viscous examples are illustrated to verify the good performance of this new finite difference US-HWENO scheme.

A fifth-order finite difference HWENO scheme combined with limiter for hyperbolic conservation laws

In this paper, a simple fifth-order finite difference Hermite WENO (HWENO) scheme combined with limiter is proposed for one- and two-dimensional hyperbolic conservation laws. The fluxes in the governing equation are approximated by the nonlinear HWENO reconstruction which is the combination of a quintic polynomial with two quadratic polynomials, where the linear weights can be artificial positive numbers only if the sum equals one. And other fluxes in the derivative equations are approximated by high-degree polynomials directly. For the purpose of controlling spurious oscillations, an HWENO limiter is applied to modify the derivatives. Instead of using the modified derivatives both in fluxes reconstruction and time discretization as in the modified HWENO scheme (Z. Zhao et al. (2020) [27]), we only apply the modified derivatives in time discretization while remaining the original derivatives in fluxes reconstruction. Comparing with the modified HWENO scheme, the proposed HWENO scheme is simpler, more accurate, efficient, and has better resolution. In addition, the HWENO scheme has a more compact spatial reconstructed stencil and is more efficient than the same order finite difference WENO scheme of Jiang and Shu (1996) [11] and WENO scheme with artificial linear weights of Zhu and Qiu (2016) [29]. Various benchmark numerical examples are presented to show the fifth-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.

High Order Asymptotic Preserving Hermite WENO Fast Sweeping Method for the Steady-State $$S_{N}$$ Transport Equations

High Order Asymptotic Preserving Hermite WENO Fast Sweeping Method for the Steady-State $$S_{N}$$ Transport Equations

An Improved Seventh Order Hermite WENO Scheme for Hyperbolic Conservation Laws

A seventh order Hermite weighted essentially non-oscillatory (HWENO) scheme [1] has been used successfully to solve hyperbolic conservation laws. However, the scheme could not achieve the optimal order of accuracy at critical points. In this paper, we propose a new central seventh order HWENO scheme. It involves seventh order HWENO reconstruction [1] and the central-upwind flux [2, 3]. For the ideal weights we used the central procedure presented in [4] and for nonlinear weights we extend the strategy proposed in [5, 6]. For time integration, the seventh order linear strong-stability-preserving Runge–Kutta (ℓSSPRK) scheme is used. The resulting scheme combines the advantages of both the improved and Hermite central schemes, e.g., compactness, improving the convergence and accuracy at critical points, decreasing the dissipation near discontinuities especially for long time evolution problems; it is simple to implement and attains seventh order accuracy. Many numerical tests are presented to validate the performance of the proposed scheme.

High order residual distribution conservative finite difference HWENO scheme for steady state problems

In this paper, we develop a high order residual distribution (RD) method for solving steady state conservation laws in a novel Hermite weighted essentially non-oscillatory (HWENO) framework recently developed in [24]. In particular, we design a high order HWENO integration for the integrals of source term and fluxes based on the point value of the solution and its spatial derivatives, and the principles of residual distribution schemes are adapted to obtain steady state solutions. Two advantages of the novel HWENO framework have been shown in [24]: first, compared with the traditional HWENO framework, the proposed method does not need to introduce additional auxiliary equations to update the derivatives of the unknown variable, and just computes them from the current point value of the solution and its old spatial derivatives, which saves the computational storage and CPU time, and thereby improves the computational efficiency of the traditional HWENO framework. Second, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller at the same grid. Thus, it is also a compact scheme when we design the higher order accuracy, compared with that in [11] Chou and Shu proposed. Extensive numerical experiments for one- and two-dimensional scalar and systems problems confirm the high order accuracy and good quality of our scheme.

A finite difference Hermite RBF‐WENO scheme for hyperbolic conservation laws

AbstractOne of the best ideas for controlling the Gibbs oscillations in hyperbolic conservation laws is to apply weighted essentially non‐oscillatory (WENO) schemes. The traditional WENO and Hermite WENO (HWENO) schemes are based on the polynomial interpolation. In this research, a non‐polynomial HENO and HWENO finite difference scheme in order to enhance the local accuracy and convergence is proposed. The idea of the reconstruction in HWENO schemes is derived from traditional WENO schemes, with the difference that in traditional WENO schemes only function values are evolved and used, while in HWENO schemes both the values of the function and the first derivative of the function are evolved in time and used. The Hermite radial basis functions (RBFs) interpolation offers the flexibility to control the local error by optimizing the free parameter. The numerical results demonstrate that the developed Hermite RBF‐ENO and RBF‐WENO schemes enhance the local accuracy and give sharper solution profile than the Hermite ENO and WENO schemes based on the polynomial interpolation.

Moment-Based Multi-Resolution HWENO Scheme for Hyperbolic Conservation Laws

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.

High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations

In this paper, we combine the nonlinear HWENO reconstruction in \cite{newhwenozq} and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in \cite{mehweno1}. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $\phi$. They are now computed from the current value of $\phi$ and the previous spatial derivatives of $\phi$. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter $\varepsilon$ used in the nonlinear weights, as observed in previous studies. Finally, in order to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods.

Well-Balanced Fifth-Order Finite Difference Hermite Weno Scheme for the Shallow Water Equations

Well-Balanced Fifth-Order Finite Difference Hermite Weno Scheme for the Shallow Water Equations

A hierarchical reconstruction for DG/FV method with low dispersion: Basic formulation and applications

In this paper, the numerical dispersion of a hierarchical reconstruction strategy for DG/FV method has been optimized. For the hierarchical strategy, the cell averages and the first derivatives are reconstructed based on the Hermite WENO method; the second derivatives are reconstructed via Green-Gauss integration and WENO reconstruction. Then, the numerical dispersion has been optimized by minimizing error with bandwidth optimization technique. Furthermore, to adjust the loss of compactness due to the reconstructed second derivatives, two modifications were also proposed with optimal weights. Eventually, from the implementations in scalar and compressible Euler equations, the superiority of numerical dispersion and accuracy of present methods could be validated. The shock capturing capacity has also been validated in 1D and 2D cases.

A compact and efficient high-order gas-kinetic scheme

In this paper, a compact fifth-order gas-kinetic scheme (GKS) is proposed to solve the Euler equations of compressible flows. It is based on the framework of a one-stage efficient high-order GKS (EHGKS), which can achieve arbitrary high-order accuracy in both space and time, and the newly-developed compact reconstruction technique, combining both the ideas of the simple weighted essentially non-oscillatory scheme (WENO) and Hermite WENO (HWENO). The evolution model of EHGKS provides the time-dependent variables at cell interfaces, which can directly compute the cell-average slope. Thus the necessary information for a fifth-order compact HWENO reconstruction technique is obtained, and also making it possible to improve simple WENO as a compact reconstruction by determining the quartic polynomial based on the compact stencil. The one-stage time discretization technique of EHGKS makes it possible to utilize the reconstruction only once in each computing time step, which is very different from the present compact multi-stage GKS schemes. The newly-developed compact reconstruction provides the polynomial, not only the pointwise variables but also their spatial derivatives for the interface variables evaluations directly. Several numerical examples are given to validate the accuracy and effectiveness of the proposed compact one-stage scheme. And the numerical results also demonstrate that the new scheme is more efficient than the original EHGKS method with WENO reconstruction technique.