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We construct a quasi-conservative alternative WENO finite difference scheme respectively coupled with the global Lax-Friedrichs (AWENO-GLF) and the contact restored Harten-Lax-van Leer approximate Riemann solver (AWENO-HLLC) for solving compressible multicomponent flows. The mass equation, the momentum equation, and the energy equation are discretized by a fully conservative AWENO-GLF or AWENO-HLLC finite difference scheme from which a consistent nonconservative discretization of the topological equation is derived according to the velocity and pressure equilibrium principle proposed by Agrall (J Comput Phys 125:150–160, 1996). We prove that, coupling with the constructed scheme, WENO interpolations with common weights for conservative variables or standard WENO interpolations with independent weights for primitive quantities can maintain velocity and pressure equilibrium. Numerical examples demonstrate that AWENO-HLLC scheme is not only less dissipative but also less oscillatory than classical WENO-GLF scheme for compressible multicomponent flows. [ABSTRACT FROM AUTHOR]

Conservation and numerical integration have been important issues for finite difference method related to robustness, reliability and accuracy requirements. In this paper, we discuss the relationship between the discretized Newton–Leibniz formula and four conservation and integration properties, including geometric conservation, flow conservation, surface integration and volume integration, for the multi-block based high-order cell-centered finite difference method. In order to achieve these conservation and integration properties, as well as multi-block compatibility, high-order accuracy, and stability within a unified methodology, we propose a new series of boundary schemes that incorporate all these constraints. To ensure geometric conservation, conservative metrics and Jacobian are adopted for coodinate transformation. To realize flow conservation, the width of the boundary stencil is enlarged to provide more degrees of freedom in order to meet the conservation constraints. To achieve uniformly high-order accuracy with arbitrary multi-block topology, cross-interface interpolation or differencing is avoided by utilizing one-sided scheme. To maintain stability, boundary interpolation scheme is designed as upwindly and compactly as possible. The proposed method is finally tested through a series of numerical cases, including a wave propagation and an isentropic vortex for accuracy verification, several acoustic tests to demonstrate the capability of handling arbitrary multi-block grid topology, a wavy channel and a closed flying wing problem for conservation verification. These numerical tests indicate that the new scheme possesses satisfactory conservation and integration properties while satisfying the requirements for high-order accuracy and stability. [ABSTRACT FROM AUTHOR]

A set of arbitrarily high-order WENO schemes for reconstructions on nonuniform grids is presented. These non-linear interpolation methods use simple smoothness indicators with a linear cost with respect to the order, making them easy to implement and computationally efficient. The theoretical analysis to verify the accuracy and the essentially non-oscillatory properties are presented together with some numerical experiments involving algebraic problems in order to validate them. Also, these general schemes are applied for the solution of conservation laws and hyperbolic systems in the context of finite volume methods. [ABSTRACT FROM AUTHOR]

In this paper, a robustness-enhanced reconstruction for the high-order finite volume scheme is constructed on the 2-D structured mesh, and both the high-order gas-kinetic scheme and the Lax-Friedrichs flux solver are considered to verify the effectiveness of this algorithm. The strategy of the successful weighted essentially non-oscillatory (WENO) reconstruction is adopted to select the smooth sub-stencils. However, there are cases where strong discontinuities exist in all sub-stencils of the WENO reconstruction, weakening its robustness. To improve the robustness of the algorithm in discontinuous regions in two-dimensional space, the hybrid reconstruction based on a combination of discontinuity feedback factor (Ji et al. in Int. J. Comput. Fluid Dyn. 35:485–509, 2021) and WENO reconstruction is developed to deal with the possible discontinuities. Numerical results from smooth to extreme cases have been presented, which validates that the new finite volume scheme is effective for robustness enhancement while maintaining high resolution compared with the WENO scheme. [ABSTRACT FROM AUTHOR]

In this study, we present a new alternative formulation of a conservative weighted essentially non-oscillatory (WENO) scheme that improves the performance of the known fifth-order alternative WENO (AWENO) schemes. In the formulation of the fifth-order AWENO scheme, the numerical flux can be written in two terms: a low-order flux and a high-order correction flux. The low-order numerical flux is constructed by a fifth-order WENO interpolator, and the high-order correction flux includes terms of the second and fourth derivatives, yielding the sixth-order truncation error. Noticing the difference in the convergence rates between these two approximations, this study first aims to fill the accuracy gap by enhancing the approximation order of the low-order numerical flux. To this end, the WENO interpolator for the low-order term is implemented using exponential polynomials with a shape parameter. Selecting a locally optimized shape parameter, the proposed WENO interpolator achieves an additional order of improvement, resulting in the overall sixth order of accuracy of the final reconstruction, under the same fifth-order AWENO framework. In addition, since a linear approximation to the high-order correction term may cause some oscillations in the vicinity of strong shocks, we present a new strategy for the limiting procedure to deal with the second derivative term in the high-order correction flux. Several numerical results for the well-known benchmark test problems confirm the reliability of our AWENO method. [ABSTRACT FROM AUTHOR]

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. 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. [ABSTRACT FROM AUTHOR]

Fast sweeping methods are a class of efficient iterative methods developed in the literature to solve steady-state solutions of hyperbolic partial differential equations (PDEs). In Zhang et al. (J Sci Comput 29:25–56, 2006) and Xiong et al. (J Sci Comput 45:514–536, 2010), high order accuracy fast sweeping schemes based on classical weighted essentially non-oscillatory (WENO) local solvers were developed for solving static Hamilton–Jacobi equations. However, since high order classical WENO methods (e.g., fifth order and above) often suffer from difficulties in their convergence to steady-state solutions, iteration residues of high order fast sweeping schemes with these local solvers may hang at a level far above round-off errors even after many iterations. This issue makes it difficult to determine the convergence criterion for the high order fast sweeping methods and challenging to apply the methods to complex problems. Motivated by the recent work on absolutely convergent fast sweeping method for steady-state solutions of hyperbolic conservation laws in Li et al. (J Comput Phys 443:110516, 2021), in this paper we develop high order fast sweeping methods with multi-resolution WENO local solvers for solving Eikonal equations, an important class of static Hamilton–Jacobi equations. Based on such kind of multi-resolution WENO local solvers with unequal-sized sub-stencils, iteration residues of the designed high order fast sweeping methods can settle down to round-off errors and achieve the absolute convergence. In order to obtain high order accuracy for problems with singular source-point, we apply the factored Eikonal approach developed in the literature and solve the resulting factored Eikonal equations by the new high order WENO fast sweeping methods. Extensive numerical experiments are performed to show the accuracy, computational efficiency, and advantages of the new high order fast sweeping schemes for solving static Hamilton–Jacobi equations. [ABSTRACT FROM AUTHOR]

In this paper, a class of high-order multi-resolution central Hermite WENO (C-HWENO) schemes for solving hyperbolic conservation laws is proposed. Formulated in a central finite volume framework on staggered meshes, the methods adopt the multi-resolution HWENO reconstructions (Li et al. in J Comput Phys 446:110653, 2021; Li et al. in Commun Comput Phys 32(2): 364-400, 2022) in space and the natural continuous extension of Runge–Kutta methods in time. Based on the zeroth-order and first-order moments of the solution defined on a series of hierarchical central spatial stencils, the proposed methods are sixth-order while the C-HWENO methods by Tao et al. (J Comput Phys 318:222-251, 2016) are fifth-order in accuracy. The linear weights of such HWENO reconstructions can be any positive numbers as long as their sum equals one, which leads to much simpler implementation and better cost efficiency than the methods by Tao et al. (J Comput Phys 318:222-251, 2016). The first-order moments are modified and the HWENO reconstructions are applied in the troubled-cells, while the linear reconstructions are used for the rest. Meanwhile, our new methods have compact stencils in the reconstructions and require neither numerical fluxes nor flux splitting. Extensive one- and two-dimensional numerical examples are performed to illustrate the accuracy and high resolution of the new C-HWENO schemes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new positivity-preserving (new PP) technique for fifth-order finite volume unequal-sized WENO schemes when solving some extreme problems of compressible Euler equations on structured meshes. After spatial reconstruction in each time marching, a detective process is used to examine the positivity of density and pressure at some checking points in each target cell. If the negativity happens at one checking point, a new compression limiter is carried out to ensure that the modified polynomials can achieve the positivity of density and pressure in the whole target cell instead of only at some discrete points. This treatment is easily implemented, owing to a quick and simple way to overestimate the minimum and maximum values of the polynomials of density and pressure in the target cell. Some numerical experiments for classical extreme problems show that the proposed method has an advantage in computational efficiency and robustness in comparison to the classical positivity-preserving techniques (Zhang and Shu in J Comput Phys 229:8918–8934, 2010), since the CFL number of 0.6 is allowed for all tests in this paper. [ABSTRACT FROM AUTHOR]

In this paper, an oscillation-free spectral volume (OFSV) method is proposed and studied for the hyperbolic conservation laws. The numerical scheme is designed by introducing a damping term in the standard spectral volume method for the purpose of controlling spurious oscillations near discontinuities. Based on the construction of control volumes (CVs), two classes of OFSV schemes are presented. A mathematical proof is provided to show that the proposed OFSV is stable and has optimal convergence rate and some desired superconvergence properties when applied to the linear scalar equations. Both analysis and numerical experiments indicate that the damping term would not destroy the order of accuracy of the original SV scheme and can control the oscillations near discontinuities effectively. Numerical experiments are presented to demonstrate the accuracy and robustness of our scheme. [ABSTRACT FROM AUTHOR]

Simulating discontinuities has been a long-standing challenge, especially when dealing with shock waves characterized by strong nonlinear features. Despite their promise, the recently developed physics-informed neural networks (PINNs) have not yet fully demonstrated their effectiveness in handling discontinuities when compared to traditional shock-capturing methods. In this study, we reveal a paradoxical phenomenon during the training of PINNs when computing problems with strong nonlinear discontinuities. To address this issue and enhance the PINNs’ ability to capture shocks, we propose PINNs-WE (Physics-Informed Neural Networks with Equation Weight) method by introducing three novel strategies. Firstly, we attenuate the neural network’s expression locally at ‘transition points’ within the shock waves by introducing a physics-dependent weight into the governing equations. Consequently, the neural network will concentrate on training the smoother parts of the solutions. As a result, due to the compressible property, sharp discontinuities emerge, with transition points being compressed into well-trained smooth regions akin to passive particles. Secondly, we also introduce the Rankine–Hugoniot (RH) relation, which is equivalent to the weak form of the conservation laws near the discontinuity, in order to improve the shock-capturing preformance. Lastly, we construct a global physical conservation constraint to enhance the conservation properties of PINNs which is key to resolve the right position of the discontinuity. To illustrate the impact of our novel approach, we investigate the behavior of the one-dimensional Burgers’ equation, as well as the one- and two-dimensional Euler equations. In our numerical experiments, we compare our proposed PINNs-WE method with a traditional high-order weighted essential non-oscillatory (WENO) approach. The results of our study highlight the significant enhancement in discontinuity computing by the PINNs-WE method when compared to traditional PINNs. [ABSTRACT FROM AUTHOR]

Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator σ S = 1 N S - 1 ∑ j ( u ¯ j - u ¯ m) 2 , where N S is the number of mesh elements in the stencil, u ¯ j is the local function average over mesh element j, and index m gives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid Hermite weighted essentially non-oscillatory scheme is proposed for nonlinear degenerate parabolic equations which may contain discontinuity in the solution. The present scheme is constructed by applying the hybrid HWENO method based on the zero- and first-order moment with DDG flux to discrete the diffusion term in spatial direction and the third-order TVD Runge–Kutta method in temporal direction. A troubled-cell indicator is first used to identify the cells in which the discontinuity may exist, then the first-order moment in the troubled-cell is reconstructed by fifth-order HWENO scheme. To avoid spurious oscillation, the HWENO reconstruction is performed when the reconstruction stencils contain troubled-cell, otherwise linear reconstruction is performed straightforwardly. Compared with WENO schemes, the present scheme has advantages: (1) compactness, only immediate neighbor cells are used in the reconstruction procedure; (2) accuracy, the numerical errors by the present scheme are smaller than those by WENO schemes. Some benchmarks for one- and two-dimensional parabolic equations to demonstrate the high order accuracy and non-oscillatory performance of the present scheme. [ABSTRACT FROM AUTHOR]

The TENO-family schemes (Fu et al. in J Comput Phys 305: 333–359, 2016) have been demonstrated to perform well for compressible gas dynamics and turbulent flow predictions on structured meshes. However, the extension of the TENO schemes to unstructured meshes is non-trivial and challenging, particularly when the multiple design objectives are pursued simultaneously, i.e., restoring the high-order accuracy in smooth regions, retaining the low numerical dissipation for small-scale features, maintaining the sharp shock-capturing property, and featuring the good numerical robustness for high-Mach flows. In this work, a family of very-high-order (up to seventh-order accuracy) robust finite-volume TENO schemes with dual ENO-like stencil selection for unstructured meshes is proposed. The stencils include one large stencil and several small stencils. The novelty originates from a so-called dual ENO-like stencil selection strategy. Following a strong scale separation, the ENO-like stencil selection procedure with a small C T is first enforced among all the candidates such that the high-order candidate scheme on the large stencil is adopted for the final reconstruction when the local flow is smooth. If the large stencil is judged to be crossed by discontinuities, a second ENO-like stencil selection with a relatively large C T is applied to all the left small stencils and the ENO property is obtained by selecting the smooth small stencils which are not crossed by discontinuities. The smaller C T in the first stage ensures that the high-order reconstruction is restored for smooth flow scales with higher wavenumbers. On the other hand, the larger C T in the second stage can enforce a strong nonlinear adaptation for capturing discontinuities with better robustness. Such a dual ENO-like stencil selection strategy introduces an explicit scale separation and deploys the optimal strategy for different types of flows correspondingly. Without parameter tuning, a set of benchmark simulations has been conducted to validate the performance of the proposed TENO schemes. Numerical results demonstrate the good numerical robustness and the low-dissipation property for highly compressible flows with shockwaves. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method’s capability using the implicit gradient method for compressible flows. [ABSTRACT FROM AUTHOR]

In this paper, a new class of high-order fast multi-resolution essentially non-oscillatory (FMRENO) schemes is proposed with an emphasis on both the performance and the computational efficiency. First, a new candidate stencil arrangement is developed for a multi-resolution representation of the local flow scales. A set of candidate stencils ranging from high- to low-order (from large to small stencils) is constructed in a hierarchical manner. Second, the monotonicity-preserving (MP) limiter is introduced as the regularity criterion of the candidate stencils. A candidate stencil, with which the reconstructed cell interface flux locates within the MP lower and upper bounds, is regarded to be smooth. Third, a multi-resolution stencil selection strategy, which prioritizes the stencils with better spectral property or higher-order accuracy meanwhile satisfying the MP criterion, is proposed. If all the candidate stencils are judged to be nonsmooth, the targeted stencil that violates the MP criterion the least is deployed as the final reconstruction instead. With this new framework, the desirable high-order accuracy is restored in the smooth regions while the sharp shock-capturing capability is achieved by selecting the targeted stencil satisfying the MP criterion most. Moreover, the new FMRENO schemes feature low numerical dissipation for resolving the broadband physical fluctuations by adaptively choosing the candidate stencil with better spectra or higher accuracy order based on the local flow regularity. Compared to the standard weighted/targeted essentially non-oscillatory (W/TENO) schemes, the computational efficiency is dramatically enhanced by avoiding the expensive evaluations of the classical smoothness indicators. A set of benchmark simulations demonstrate the performance of the new FMRENO schemes for handling complex fluid problems with a wide range of length scales. [ABSTRACT FROM AUTHOR]

The finite difference multi-resolution alternative weighted essentially non-oscillatory (MR-AWENO) scheme has been designed to solve hyperbolic conservation laws (Wang et al. in Comput Methods Appl Mech Eng 382:113853, 2021). However, the scheme is not scale-invariant and generates numerical oscillations near strong shocks. To overcome this issue, we design the scale-invariant Si-weights and the MR-AWENO-Si operator by including the new global smoothness indicator and the descaler. The resulting scale-invariant MR-AWENO-Si scheme captures discontinuity of any scale in the essentially non-oscillatory (ENO) way efficiently and robustly. We also demonstrate an interesting application of the scale-invariant scheme to achieve the well-balanced property for the compressible Euler equations under a gravitational potential field by modifying the numerical fluxes, reformulating the source terms, and enforcing the MR-AWENO-Si operator. A theoretical proof is given, and extensive one- and two-dimensional classical examples are used to verify the performance of the MR-AWENO-Si scheme in terms of accuracy, robustness, and well-balanced property. [ABSTRACT FROM AUTHOR]

A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, and closed manifolds (surface PDEs or SPDEs in R 2 and R 3 ). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers’ equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers’ equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers’ equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds. [ABSTRACT FROM AUTHOR]

In this paper, a fifth-order finite difference mapped weighted essentially non-oscillatory scheme with unequal-size stencils, termed as MUS-WENO, is designed for hyperbolic conservation laws. The new mapping function and nonlinear weights are proposed to reduce the difference between the linear weights and nonlinear weights. It could get smaller numerical errors and obtain fifth-order accuracy, give optimal fifth-order convergence with a tiny ε even near critical points in smooth regions, and simultaneously suppress spurious oscillations near strong discontinuities. Comparing with the classical fifth-order finite difference WENO schemes and fifth-order finite difference mapped WENO (MWENO) schemes, this mapped WENO scheme uses three unequal-size stencils and the linear weights which can be any positive numbers on condition that their summation is one. Another advantage is that we can reconstruct polynomial over the whole big stencil, while many classical high-order WENO reconstructions only reconstruct the values at the boundary points or discrete quadrature points. Extensive examples including some steady-state problems and some low density, low pressure, or low energy extreme problems are used to testify the good representations of this new MUS-WENO scheme. [ABSTRACT FROM AUTHOR]

Adaptive mesh refinement (AMR) and wavelet-based multi-resolution technique, which refine the spatial resolution in regions of interest and coarsen the mesh in other regions, are widely used in scientific computing for higher computational efficiency. The key component of AMR and the multi-resolution method is the adaptation strategy including the regularity estimation. In this paper, a new adaptation strategy for AMR and the multi-resolution method is proposed. Different from AMR, which measures the function regularity based on an empirical gradient operator, and the multi-resolution method with complicated wavelet analysis, the new approach examines the solution smoothness based on the high-order TENO reconstruction (Fu et al. in J Comput Phys 305:333–359, 2016). The core idea is that the TENO scheme not only provides the reconstructed data at the cell interface for flux evaluation but also classifies the local flow scales as smooth or nonsmooth on the discretized mesh. Since the scale-separation procedure is achieved in the spectral and characteristic space, the new adaptation strategy is weakly problem-dependent. Moreover, the extra complexity due to the empirical gradient computing or the wavelet analysis is eliminated. Based on Harten’s finite-volume multi-resolution method (Harten in J Comput Phys 115(2):319–338, 1994), a set of benchmark cases is simulated to validate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

The recently proposed high-order TENO scheme [Fu et al., Journal of Computational Physics, 305(2016): 333-359] has shown great potential in predicting complex fluids owing to the novel weighting strategy, which ensures the high-order accuracy, the low numerical dissipation, and the sharp shock-capturing capability. However, the applications are still restricted to simple geometries with Cartesian or curvilinear meshes. In this work, a new class of high-order shock-capturing TENO schemes for unstructured meshes are proposed. Similar to the standard TENO schemes and some variants of WENO schemes, the candidate stencils include one large stencil and several small third-order stencils. Following a strong scale-separation procedure, a tailored novel ENO-like stencil selection strategy is proposed such that the high-order accuracy is restored in smooth regions by selecting the candidate reconstruction on the large stencil while the ENO property is enforced near discontinuities by adopting the candidate reconstruction from smooth small stencils. The nonsmooth stencils containing genuine discontinuities are explicitly excluded from the final reconstruction, leading to excellent numerical stability. Different from the WENO concept, such unique sharp stencil selection retains the low numerical dissipation without sacrificing the shock-capturing capability. The newly proposed framework enables arbitrarily high-order TENO reconstructions on unstructured meshes. For conceptual verification, the TENO schemes with third- to sixth-order accuracy are constructed. Without parameter tuning case by case, the performance of the proposed TENO schemes is demonstrated by examining a set of benchmark cases with broadband flow length scales. [ABSTRACT FROM AUTHOR]

In this paper, based on our previous optimal compact schemes with minimized dispersion and controllable dissipation (OC–WENO schemes) (Sun et al. in Sci China-Phys Mech Astron 57:971–982, 2014), a spatio-temporal optimized, hybird compact-WENO scheme with minimized dispersion and critical-adaptive dissipation is developed for solving compressible flows. Firstly, the spectral properties of the fourth-order OC–WENO scheme is researched within the spatio-temporal discrete framework. In conjunction with total dispersion error of the fully scheme, an integrated error function is designed to optimize the dispersion property. Secondly, the scale sensor leveraged to quantify the local scaled wavenumber is optimized in the wavenumber space to improve the accuracy of estimating. Moreover, a dispersion-dissipation condition, controlling the relative proportion of dispersion and dissipation errors, is developed for the fully discrete scheme. Thirdly, by exploiting the optimized scale sensor and the dispersion-dissipation condition, the critical-adaptive dissipation surface is constructed to achieve the adaptive dissipation property relating to the local characteristics of the flow fields and different Courant number. To have the shock-capturing capability, the proposed compact scheme is blended with fifth-order WENO scheme to form the MDADFC–WENO scheme. Finally, a set of benchmark test cases is employed to validate the good performance of the proposed scheme. [ABSTRACT FROM AUTHOR]

Fu et al. (J Comput Phys 374:724–751, 2018), a class of adaptive targeted ENO (TENO) schemes have been proposed. Excellent resolution and robustness of the TENO scheme were validated by benchmark problems, but the computational cost is high. In order to improve the computational efficiency, a modified TENO scheme is proposed in this article. First, using the weighting strategy of the fifth-order TENO as an explicit discontinuity detector, locations of discontinuities are detected on the five-point stencil, and the initial target stencil with maximum support on the five-point stencil is obtained. Then, with a simple discontinuity detection strategy, the initial target stencil is enlarged point by point in the direction where no discontinuity is detected. In this manner, final target stencil with maximum support, not crossed by discontinuities, on the full six- or eight-point stencil is formed. Unlike TENO using candidate stencils with incrementally increased width, all stencils are three-point in the new framework. Due to the simplified determination of final target stencil and avoiding expensive calculation of smoothness indicator of large stencil, the computational cost of TENO-M is significantly lower than that of TENO. Several benchmark problems are conducted to validate the performance of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, an efficient low-dissipation high-order TENO scheme is proposed for ideal MHD flows. For high computational efficiency, a troubled-cell indicator based on the ENO-like stencil selection strategy in TENO schemes is introduced to isolate the discontinuities from smooth regions. While the high-order linear scheme is adopted for the smooth regions, a low-dissipation TENO scheme is applied for capturing discontinuities detected by the troubled-cell indicator. The case-independent parameters are given based on spectral analysis. Both the governing equations of the ideal MHD and the Hamilton–Jacobi type constrained transport equation for divergence-free condition can be solved by the newly proposed scheme. Since most computational regions are resolved by the linear scheme without expensive characteristic decomposition, flux splitting and nonlinear weight calculation, the proposed scheme is highly efficient. A set of benchmark cases has been simulated to demonstrate the performance of the proposed scheme. Numerical results reveal that remarkable speedup is achieved by the present scheme while the oscillation-free property and the high-order accuracy are preserved. [ABSTRACT FROM AUTHOR]

A new seventh-order weighted compact adaptive scheme is proposed in this paper. The proposed reconstruction is inspired by the upwind compact scheme and the adaptive order weighted essentially non-oscillatory (WENO) scheme. The proposed seventh-order scheme is a convex combination of a linear seventh-order compact scheme and three linear third-order compact schemes. The reconstruction of the proposed scheme is based on the same stencil as the reconstruction of the original fifth-order weighted compact scheme. Various classical tests are presented to show the performance of the proposed numerical scheme. The numerical results of the proposed scheme are compared with those obtained with the original fifth-order compact WENO scheme, seventh-order compact WENO scheme and classical WENO schemes. The numerical results show that the proposed scheme preserves the designed seventh-order of accuracy in smooth regions and produces higher resolution solutions near shocks or discontinuities without numerical oscillations with less computational cost than the original weighted compact schemes. [ABSTRACT FROM AUTHOR]

A fifth order finite difference alternative weighted essentially non-oscillatory scheme is studied for a five-equation model, which plays an important role in the modelling of compressible multi-component flows. In our algorithm, the primitive variables are used in the weighted essentially non-oscillatory interpolation, from which it can be proved that the equilibriums of the velocity and pressure are preserved respectively throughout the whole computation of contact moving interface problems, under the ideal and stiffened gases equations of state. In addition, it is found that with the interpolation being performed on the characteristic variables which are the projections of the primitive variables, non-physical oscillations appearing around the contact waves can be significantly restrained. Numerical examples verify the theory and illustrate the robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

For the simulation of flow fields with a broad range of length scales, designing numerical schemes with good spectral properties is one of the most important issues. To improve the spectral properties of the semi-discrete finite different schemes, the authors have previously proposed the idea of optimizing the dispersion and dissipation properties separately and a class of finite difference scheme with minimized dispersion and controllable dissipation properties is thus developed (Sun et al. J Comput Phys 230:4616–4635, 2011, 270:238–254, 2014). In the present paper, we further investigated this idea and extend it to the fully discrete scheme. In other words, the dispersion and dissipation errors induced by the temporal discretization are taken into consideration in the present paper. Moreover, a scale sensor is designed in the control of dissipation error to ensure the numerical scheme can automatically adjust its dissipation according to the local characteristic of the flow field. To achieve the shock-capturing capability, this optimized scheme is blended with the WENO-Z scheme to form a hybrid scheme. A number of benchmark test cases including the transportation of a linear wave, the propagation of a sound wave packet, the Shu–Osher problem, the double Mach reflection problem and the Rayleigh–Taylor instability problem are employed to verify the good spectral properties and robust shock-capturing capability of the proposed scheme. [ABSTRACT FROM AUTHOR]

When constructing high-order schemes for solving hyperbolic conservation laws, the corresponding high-order reconstructions are commonly performed in characteristic spaces to eliminate spurious oscillations as much as possible. For multi-dimensional finite volume (FV) schemes, we need to perform the characteristic decomposition several times in different normal directions of the target cell, which is very time-consuming. In this paper, we propose a rotated characteristic decomposition technique which requires only one-time decomposition for multi-dimensional reconstructions. The rotated direction depends only on the gradient of a specific physical quantity which is cheap to calculate. This technique not only reduces the computational cost remarkably, but also controls spurious oscillations effectively. We take a third-order weighted essentially non-oscillatory finite volume (WENO-FV) scheme for solving the Euler equations as an example to demonstrate the efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]

On the idea of mapped WENO scheme, properties of mapping methods are analyzed, uncertainties in mapping development are investigated, and new piecewise rational mappings are proposed. Based on our former understandings, i.e. the mapping at endpoints {0, 1} tending to identity mapping, a so-called Cn,m condition is summarized for function development. Uncertainties, i.e., whether the pattern at endpoints of mapping would make mapped scheme behave like WENO or ENO, whether piecewise implementation of mapping would entail numerical instability, and whether WENO3 could preserve the third-order at first-order critical points by mapping, are analyzed and clarified. A new piecewise rational mapping with sufficient regulation capability is developed afterwards, where the flatness of mapping around the linear weights and the profile at endpoint tending toward identity mapping can be coordinated explicitly and simultaneously. Hence, the increase of resolution and preservation of stability can be balanced. Especially, concrete mappings are determined for {WENO3, 5, 7}. Numerical examples are tested for the new mapped WENO, which regard preservation and convergence rate of accuracy, numerical stability including that in the long-time computation, resolution and robustness. For comparison, some recent mappings such as IM by [App. Math. Comput. 232, 2014:453–468], RM by [J. Sci. Comput. 67, 2016:540–580] and AIM by [J. Comput. Phys. 381, 2019:162–188] are tested; in addition, some recent WENO-Z type schemes are chosen as well. The results manifest that new schemes can preserve optimal orders at corresponding critical points, achieve numerical stability, and indicate overall comparative advantages regarding accuracy, resolution and robustness. [ABSTRACT FROM AUTHOR]

The MHLLE (Multidimensional Harten, Lax, van Leer and Einfeldt) scheme, which is a genuinely multidimensional Riemann solver proposed by Balsara, encounters the accuracy problem at low speeds because it is built upon the compressible Euler equations. In order to overcome this problem, asymptotic analysis on the MHLLE scheme is conducted in this study. Based on the asymptotic analysis, a novel multidimensional Riemann solver called MHLLELS (Multidimensional Harten, Lax, van Leer and Einfeldt scheme for Low Speeds) for curvilinear coordinates is proposed. Systematic numerical cases, including 2d inviscid NACA (National Advisory Committee for Aeronautics) 0012 airfoil, Gresho vortex problem, separated flows around a circular cylinder (M∞ = 0.01), turbulent flow over a flat plate, turbulent flow past a NACA0012 airfoil, turbulent flow past a backward facing step, and spherical blast wave, are carried out. Results indicate that the MHLLELS scheme proposed in this study improves the MHLLE scheme’s accuracy at low speeds remarkably, while it is with a high resolution in multidimensional cases. It is promising to be widely used in both scholar and engineering areas. [ABSTRACT FROM AUTHOR]

Euler equations with a gravitational source term (PDEs) admit a hydrostatic equilibrium state where the source term exactly balances the flux gradient. The property of exact preservation of the equilibria is highly desirable when the PDEs are numerically solved. Li and Xing (J Comput Phys 316:145–163, 2016) proposed a high-order well-balanced characteristic-wise finite volume weighted essentially non-oscillatory (FV-WENO) scheme for the cases of isothermal equilibrium and polytropic equilibrium. On the contrary to what was claimed, the scheme is not well-balanced. The root of the problem is the precarious effects of a non-zero sensitivity parameter in the nonlinear weights of the WENO polynomial reconstruction procedure (WENO operator). The effects are identified in the theoretical proof for the well-balanced scheme and verified numerically on a coarse mesh resolution and a long time simulation of the PDEs. In this study, two simple yet effective numerical techniques derived from the multiplicative-invariance (MI) property of a WENO operator are invoked to rectify the sensitivity parameter’s dependency yielding a correct proof for the sensitivity parameter-independent (characteristic-wise) well-balanced FV-WENO scheme. The (non-)well-balanced nature of the schemes is demonstrated with several one- and two-dimensional benchmark steady state problems and a small perturbation over the steady state problems. Moreover, the one-dimensional Sod problem under the gravitational field is also simulated for showing the performance of the well-balanced FV-WENO scheme in capturing shock, contact discontinuity, and rarefaction wave in an essentially non-oscillatory nature. It also indicates that the numerical scheme with the third-order Runge–Kutta time-stepping scheme should take the CFL number less than 0.5 to mitigate the Gibbs oscillations at the shock without increasing the numerical dissipation artificially in the Lax–Friedrichs numerical flux. [ABSTRACT FROM AUTHOR]

High order methods play important roles in the modelling of compressible multi-component flows. However, they may generate negative sound speed, which leads to an instability of the numerical schemes. In this paper, we propose bound- and positivity-preserving limiters for high order finite difference schemes, based on which the equilibriums of the velocity and pressure are preserved throughout the whole computation of contact moving interface problems with the ideal and stiffened equations of state. For illustration purpose, high order alternative WENO scheme is taken for example. Numerical examples verify the theory and demonstrate the robustness of the proposed bound- and positivity-preserving limiters. [ABSTRACT FROM AUTHOR]

To enhance the performance of the third-order weighted essentially non-oscillatory (WENO) scheme on the four-point stencil, a new hybrid central-type scheme is proposed in this paper. Developments are conducted in the following two aspects. First, a new fourth-order central WENO scheme, named as WENO4-LC, is devised for shock capturing, and a low dissipative third-order grid-centered upwind scheme is adopted for the smooth region. Second, a discontinuity indicator is extended to the four-point stencil version, which is simpler than its counterpart in our former article (Guo et al. in J Sci Comput 83, 28, 2020), for switching between the linear and nonlinear branch of the hybrid scheme. Numerical tests of the Euler and Navier–Stokes benchmarks show that the new indicator is good on detecting the discontinuities. Moreover, the performances of the central-type WENO4-LC and its corresponding hybrid scheme have obvious improvements compared with those of WENO3-JS and WENO3-L, and even behave slightly better than classical WENO5-JS on resolving the fluid structures. Meanwhile, the hybrid scheme is efficient which only costs 76% CPU time of WENO3-JS and about 58% of WENO5-JS. [ABSTRACT FROM AUTHOR]

Screech tones are the high intensity shock-associated noise with discrete frequency in imperfectly supersonic jet. Accuracte numerical simulation of shock-associated noise requires numerical scheme with high order accuracy, low dissipation and low dispersion as well as robust shock-capturing ability. The applicability of eight kinds of upwind/symmetric WENO schemes for the direct numerical simulation of screech tone is evaluated through comparison study on the spectral characteristics and direct numerical simulations of underexpanded supersonic cold jet issuing from circular sonic nozzle. The spectral characteristics based on the approximate dispersion relation shows that the adaptive central-upwind (WENO-SYMCU-6) scheme has the best resolution and the minimum value of point-per-wavelength. However, the comparison of nonlinear response indicates that the improved upwind WENO-ZM-5 ( p = 1 ) scheme has almost the weakest nonlinear response for the single Fourier mode. And the performances in the direct numerical simulations also show that the WENO-ZM-5 ( p = 1 ) scheme is the best. With this method, not only the screech tones of the high amplitude axisymmetric mode are obtained, but also a variety of high-frequency coupling modes with the highest amplitudes can be distinguished in a wider spectrum range. And there is no nonphysical high-frequency wave induced by the nonlinear implementation of WENO schemes at the near region of the nozzle exit lip. [ABSTRACT FROM AUTHOR]

A new approach is proposed to detect edges based on an artificial neural network (ANN). Some elementary continuous and discontinuous functions interpolated in the polynomial space and their continuity are used as the training sets to train a back propagation neural network containing two hidden layers. The ANN edge detector is used to detect the edges in an image and the locations of discontinuity in the hybrid fifth order Compact-WENO nonlinear (Hybrid) scheme for solving hyperbolic conservation laws with solutions containing both discontinuous and complex fine scale structures. Several classical examples in the image processing show that the ANN edge detector can capture an edge accurately with fewer grid points than the classical multi-resolution analysis. Furthermore, as oppose to the MR analysis, the ANN edge detector is robust with no problem dependent parameter, in addition to being accurate and efficient. The performance of the Hybrid scheme with the ANN edge detector is demonstrated with several one- and two-dimensional benchmark examples in the shallow water equations and Euler equations. [ABSTRACT FROM AUTHOR]

In the present work, we propose a new discontinuity indicator for constructing shock-capturing finite difference schemes based on the self-adaptation algorithm in the WENO-OS4 scheme (Li et al. in J Sci Comput 71:109–143, 2017). The new discontinuity indicator can identify the discontinuities as accurate as the original self-adaptation algorithm through using a more straightforward numerical condition. The high-resolution and the problem-independency of the original algorithm for identifying discontinuities are maintained, but the excessive expenses are greatly reduced by the decoupled implementation and the employment of the local characteristic projection. Moreover, new hybrid WENO schemes are constructed based on this new discontinuity indicator. The hybrids are carried out between nonlinear WENO schemes with shock-capturing ability and linear upwind or central schemes with high resolution. Both the effectiveness of the indicator and the good resolution of the hybrid schemes are demonstrated by the numerical tests. The results of the tests indicate that the new hybrid WENO schemes have significant improvements on both resolution and computational efficiency when compared with the WENO-JS5 and WENO-OS4. [ABSTRACT FROM AUTHOR]

Including polynomials with small degree and stencil when designing very high order reconstructions is surely beneficial for their non oscillatory properties, but may bring loss of accuracy on smooth data unless special care is exerted. In this paper we address this issue with a new Central WENOZ (CWENOZ ) approach, in which the reconstruction polynomial is computed from a single set of non linear weights, but the linear weights of the polynomials with very low degree (compared to the final desired accuracy) are infinitesimal with respect to the grid size. After proving general results that guide the choice of the CWENOZ parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order WENO reconstructions (Arbogast et al. in SIAM J Numer Anal 56(3):1818-1947, 2018),(Balsara et al. in J Comput Phys 326:780-804, 2016). The novel reconstruction yields similar accuracy and oscillations with respect to the previous ones, but saves up to 20% computational time since it does not rely on a hierarchic approach and thus does not compute multiple sets of nonlinear weights in each cell. [ABSTRACT FROM AUTHOR]

In this paper, new finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving the Hamilton–Jacobi equations on structured meshes. The crucial idea of the spatial reconstructions is borrowed from the original HWENO schemes (Qiu and Shu in J Comput Phys 204:82–99, 2005), in which the function and its first derivative values are evolved in time and used in the reconstruction. Such new HWENO spatial reconstructions with the application of three unequal-sized spatial stencils result in an important innovation that we perform only spatial HWENO reconstructions for numerical fluxes of function values and high-order linear reconstructions for numerical fluxes of derivatives, which are different to other HWENO schemes. The new HWENO schemes could obtain smaller errors with optimal high-order accuracy in smooth regions, and keep sharp transitions and non-oscillatory property near discontinuities. Extensive benchmark examples are performed to illustrate the good performance of such new finite difference HWENO schemes. [ABSTRACT FROM AUTHOR]

The aim of this study is to present an improved third-order weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws. We first present a novel smoothness indicator by using discrete differential operator which annihilates exponential polynomials. The new smoothness indicator can vanish to zero in smooth regions with higher rates than the classical methods such that it can distinguish the smooth region and discontinuity more efficiently. The proposed scheme achieves the maximal approximation order without loss of accuracy at critical points. A detailed analysis is provided to verify the third-order accuracy. The proposed scheme attains better resolution in smooth regions, while reducing numerical dissipation significantly near singularities. The advantages are more pronounced in two-dimensional model problems. Some numerical experiments are provided to illustrate the performance of the proposed WENO scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel numerical scheme for solving a class of nonlinear degenerate parabolic equations with non-smooth solutions. The proposed method relies on a special kernel based formulation of the solutions found in our early work on the method of lines transpose and successive convolution. In such a framework, a high order weighted essentially non-oscillatory methodology and a nonlinear filter are further employed to avoid spurious oscillations. High order accuracy in time is realized by using the high order explicit strong-stability-preserving (SSP) Runge-Kutta method. Moreover, theoretical investigations of the kernel based formulation combined with an explicit SSP method indicate that the combined scheme is unconditionally stable and up to third order accuracy. Evaluation of the kernel based approach is done with a fast O (N) summation algorithm. The new method allows for much larger time step evolution compared with other explicit schemes with the same order accuracy, leading to remarkable computational savings. [ABSTRACT FROM AUTHOR]

The fifth, seventh and ninth order characteristic-wise alternative weighted essentially non-oscillatory (AWENO) finite difference schemes are applied to the fully conservative (FC) form and the overestimated quasi-conservative (OQC) form of the compressible multicomponent flows. Several linear and nonlinear numerical operators such as the linear Lax–Friedrichs operator and linearized nonlinear WENO operator and their mathematical properties are defined in order to build a general mathematical (numerical) framework for identifying the necessary and sufficient conditions required in maintaining the equilibriums of certain physical relevant properties discretely. In the case of OQC form, the AWENO scheme with the modified flux can be rigorously proved to maintain the equilibriums of velocity, pressure and temperature. Furthermore, we also show that the FC form cannot maintain the equilibriums without an additional advection equation of auxiliary variable involving the specific heat ratio. Extensive one- and two-dimensional classical benchmark problems, such as the moving material interface problem, multifluid shock-density interaction problem and shock-R22-bubble interaction problem, verify the theoretical results and also show that the AWENO schemes demonstrate less dissipation error and higher resolution than the classical WENO-Z scheme in the splitting form (Nonomura and Fujii in J Comput Phys 340:358–388, 2017). [ABSTRACT FROM AUTHOR]

A modified fifth order Z-type (nonlinear) weights, which consist of a linear term and a nonlinear term, in the weighted essentially non-oscillatory (WENO) polynomial reconstruction procedure for the WENO-Z finite difference scheme in solving hyperbolic conservation laws is proposed. The nonlinear term is modified by a modifier function that is based on the linear combination of the local smoothness indicators. The WENO scheme with the modified Z-type weights (WENO-D) scheme and its improved version (WENO-A) scheme are proposed. They are analyzed for the maximum error and the order of accuracy for approximating the derivative of a smooth function with high order critical points, where the first few consecutive derivatives vanish. The analysis and numerical experiments show that, they achieve the optimal (fifth) order of accuracy regardless of the order of critical point with an arbitrary small sensitivity parameter, aka, satisfy the Cp-property. Furthermore, with an optimal variable sensitivity parameter, they have a quicker convergence and a significant error reduction over the WENO-Z scheme. They also achieve an improved balance between the linear term, which resolves a smooth function with the fifth order upwind central scheme, and the modified nonlinear term, which detects potential high gradients and discontinuities in a non-smooth function. The performance of the WENO schemes, in terms of resolution, essentially non-oscillatory shock capturing and efficiency, are compared by solving several one- and two-dimensional benchmark shocked flows. The results show that they perform overall as well as, if not slightly better than, the WENO-Z scheme. [ABSTRACT FROM AUTHOR]

First this paper analyzes the reason for the accuracy losing of the third-order weighted essentially non-oscillatory (WENO) scheme. It is shown that one reason is that the local smoothness indicators of the third-order WENO scheme cannot correctly treat the smooth three-point stencil containing a non-nodal critical point, here, 'non-nodal' means the critical point is not a grid point. And then a discontinuity-detecting method for a four-point stencil is proposed and applied to develop the high order accurate hybrid-WENO scheme by combining the third-order WENO scheme and a third-order upstream scheme. This four-point stencil is actually the stencil used for constructing the third-order WENO scheme (positive and negative numerical fluxes), hence the resulting hybrid-WENO scheme proposed by this paper does not introduce new grid point. Numerical examples show that the detecting method and the hybrid scheme are robust for problems with shocks, and the hybrid scheme obtains real third-order convergence rate for smooth solutions containing critical points. [ABSTRACT FROM AUTHOR]

We present a novel efficient adaptive central-upwind WENO-CU6 scheme (named as EWENO-CU6) to maximize the order of accuracy, minimize dissipation and improve numerical robustness. Unlike WENO-CU6 scheme, the present paper introduces a switch formula based on our new shock sensor to adaptively diminish numerical dissipation in a rational manner when departing from discontinuities. Based on the above improvements, the performances of several schemes are characterized in scalar equations, which indicate new scheme's high order of accuracy, robustness with a reasonable computational efficiency in wavenumber space. A variety of benchmark cases of Euler equations are tested to further verify the new scheme's performance. From multi-dimensional tests, EWENO-CU6 scheme shows a greater resolving power in practical flow cases involving both shocks and complex features. We emphasize that this method can be applied more broadly for any algorithm which contains the similar reference smoothness indicator in WENO-CU class schemes. [ABSTRACT FROM AUTHOR]

In this paper, a modified fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme is presented. The quadratic polynomial approximation of numerical flux on each candidate stencil of the traditional WENO-JS scheme is modified by adding a form of cubic terms such that the resulting stencil approximation achieves fourth-order accuracy. And the corresponding smoothness indicators are calculated. The modified candidate fluxes and local smoothness indicators, when used in the WENO-JS scheme, can make the resulting new scheme (called WENO-MS) achieve fifth-order convergence in smooth regions including first-order critical points. A series of one- and two-dimensional numerical examples are presented to demonstrate the performance of the new scheme. The numerical results show that the proposed WENO-MS scheme provides a comparable or higher resolution of fine structures compared with the WENO-M, WENO-Z and P-WENO schemes, while it increases only 7% of CPU time compared with the traditional WENO-JS scheme. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid method suitable for solving the Euler equations using high order methods has been proposed. The method was implemented and validated with a seventh order WENO scheme in OpenFOAM®. The hybrid method combines a simple MUSCL-type flux approach and a characteristic flux approach. In the MUSCL-type flux approach, the inviscid fluxes are computed using approximate Riemann solvers HLL and HLLC schemes based on the WENO-reconstructed state variables. Hence, this is dubbed as the VF (variable-based flux) approach. In critical regions where VF may produce spurious oscillations, a novel, low-dissipation HLL-based CF (characteristic flux) approach is applied. Critical regions were identified using a modified Bhagatwala–Lele shock sensor. The VF/CF hybrid method has been shown to produce high-resolution, essentially non-oscillatory results for a number of 1D and 2D problems at a fraction of the cost of a pure CF approach. Moreover, a 2D advection problem was designed to investigate the choice of state variables and flux schemes. The results have shed more light on the relation between Kelvin–Helmholtz roll-ups and numerical instabilities along slip lines. [ABSTRACT FROM AUTHOR]

In this paper, we propose a high order residual distribution conservative finite difference scheme for solving steady state conservation laws. A new type of WENO (weighted essentially non-oscillatory) termed as WENO-ZQ integration is used to compute the numerical fluxes and source term based on the point values of the solution, and the principles of residual distribution schemes are adapted to obtain steady state solutions. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving shocks of the proposed methods. [ABSTRACT FROM AUTHOR]

A new hybrid optimized low-dissipation and adaptive MUSCL scheme is present for finite volume method. The proposed scheme, based on an optimized linear scheme with monotonicity preserving limitation and an adaptive MUSCL scheme, emphasizes on the resolution rather than the formal order. This technique is applied to cell interface reconstruction and bears similarity in form to the widely used MUSCL scheme except wider interpolation stencil. Although the scheme is not high order of accuracy because of adaptive MUSCL scheme acting as nonlinear part, the low-dissipation feature from the optimized linear part makes it very accurate and robust in many practical applications, where much richer flow structures can be obtained. A number of test cases are solved to validate the high resolution of the present scheme. [ABSTRACT FROM AUTHOR]

In this study, we present a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving Hamilton-Jacobi equations. The proposed scheme recovers the maximal approximation order in smooth regions without loss of accuracy at critical points. We incorporate exponential polynomials into the scheme to obtain better approximation near steep gradients without spurious oscillations. In order to design nonlinear weights based on exponential polynomials, we suggest an alternative approach to construct Lagrange-type exponential functions reproducing the cell-average values of exponential basis functions. Using the Lagrange-type exponential functions, we provide a detailed analysis of the approximation order of the proposed WENO scheme. Compared to other WENO schemes, the proposed scheme is simpler to implement, yielding better approximations with lower computational costs. A number of numerical experiments are presented to demonstrate the performance of the proposed scheme. [ABSTRACT FROM AUTHOR]