Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes

In this paper, we introduce a new type of troubled-cell indicator to improvehybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-orderlinear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore,they can reduce computational cost while maintaining non-oscillatory properties innon-smooth regions. Reliable troubled-cell indicators are essential for efficient hybridWENO methods. Most of troubled-cell indicators require proper parameters to detectdiscontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theoremthat does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations areconducted to demonstrate the performance of the proposed indicator. The results withthe proposed troubled-cell indicator are in good agreement with pure WENO schemes.Also the new indicator has advantages in the computational cost compared with theother indicators.     

Convolution Neural Network Shock Detector for Numerical Solution of Conservation Laws

We propose a universal discontinuity detector using convolution neural network (CNN) and apply it in conjunction of solving nonlinear conservation laws in both1D and 2D. The CNN detector is trained offline with synthetic data. The training dataare generated using randomly constructed piecewise functions, which are then processed using randomized linear advection solver to count for the cases of numericalerrors in practice. The detector is then paired with high-order numerical solvers. Inparticular, we combined high-order WENO in troubled cells with high-order centraldifference in smooth region. Extensive numerical examples are presented. We observethat the proposed method produces notably sharper and cleaner signals near the discontinuities, when compared to other well known troubled cell detector methods.     

A Five-Point TENO Scheme with Adaptive Dissipation Based on a New Scale Sensor

In this paper, a new five-point targeted essentially non-oscillatory (TENO)scheme with adaptive dissipation is proposed. With the standard TENO weightingstrategy, the cut-off parameter $C_T$ determines the nonlinear numerical dissipation ofthe resultant TENO scheme. Moreover, according to the dissipation-adaptive TENO5-A scheme, the choice of the cut-off parameter $C_T$ highly depends on the effective scalesensor. However, the scale sensor in TENO5-A can only roughly detect the discontinuity locations instead of evaluating the local flow wavenumber as desired. In thiswork, a new five-point scale sensor, which can estimate the local flow wavenumber accurately, is proposed to further improve the performance of TENO5-A. In combinationwith a hyperbolic tangent function, the new scale sensor is deployed to the TENO5-Aframework for adapting the cut-off parameter $C_T,$ i.e., the local nonlinear dissipation,according to the local flow wavenumber. Overall, sufficient numerical dissipation isgenerated to capture discontinuities, whereas a minimum amount of dissipation is delivered for better resolving the smooth flows. A set of benchmark cases is simulated todemonstrate the performance of the new TENO5-A scheme.     

A Modified Adaptive Improved Mapped WENO Method

We propose a new family of mapped WENO schemes by using several adaptive control functions and a smoothing approximation of the signum function. Theproposed schemes admit an extensive permitted range of the parameters in the mapping functions. Consequently, they have the capacity to achieve optimal convergencerates, even near critical points. Particularly, the new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. Itproduces numerical solutions with high resolution without generating spurious oscillations, especially for long output times.     

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

In this paper, a new multi-resolution weighted essentially non-oscillatory(MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENOlimiter is a new extension of the finite volume MR-WENO schemes. Such new limiteruses information of the LDG solution essentially only within the troubled cell itself, tobuild a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to thehighest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which couldmaintain the original order of accuracy in smooth regions and could simultaneouslysuppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on conditionthat their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify thefreedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiteris very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstructionmethodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classicalviscous examples are given to show the good performance of this third-order LDGmethod with associated MR-WENO limiter.     

Multi-Resolution Method Based on Riemann Solvers for Detonation and Deflagration in High Dimension

In this paper, we propose accurate Riemann solvers for detonation and deflagration with sharp interface in high dimension. The standard finite volume scheme is used for each fluid away from material interface, the detonation and the deflagration interfaces are captured by the level set method, small cut cells are treated with a mixing procedure to get stable algorithm. By Riemann solver for the detonation and the deflagration, the interface fluxes are obtained. With the help of the adaptive multi-resolution algorithms, we extend the method to three dimension conveniently. Numerical examples in two or three-dimension are carried out to demonstrate the potential and robustness of the method.     

Adaptive Multi-Resolution Method for 3D Reactive Flows with Level Set Front Capturing

For compressible reactive flows with stiff source terms, a new block-based adaptive multi-resolution method coupled with the adaptive multi-resolution representation model for ZND detonation and a conservative front capturing method based on a level-set technique is presented. When simulating stiff reactive flows, underresolution in space and time can lead to incorrect propagation speeds of discontinuities, and numerical dissipation makes it impossible for traditional shock-capturing methods to locate the detonation front. To solve these challenges, the proposed method leverages an adaptive multi-resolution representation model to separate the scales of the reaction from those of fluid dynamics, achieving both high-resolution solutions and high efficiency. A level set technique is used to capture the detonation front sharply and reduce errors due to the inaccurate prediction of detonation speed. In order to ensure conservation, a conservative modified finite volume scheme is implemented, and the front transition fluxes are calculated by considering a Riemann problem. A series of numerical examples of stiff detonation simulations are performed to illustrate that the present method can acquire the correct propagation speed and accurately capture the sharp detonation front. Comparative numerical results also validate the approach’s benefits and excellent performance.     

Performance Enhancement for High-Order Gas-Kinetic Scheme Based on WENO-Adaptive-Order Reconstruction

High-order gas-kinetic scheme (HGKS) has been well-developed in the past years. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness. However, there is still room for its further improvement. Firstly, the reconstruction in the previous scheme mainly achieves a fifth-order accuracy for the point-wise values at a cell interface due to the use of standard WENO reconstruction, and the slopes of the initial non-equilibrium states have to be reconstructed from the cell interface values and cell averages again. The same order of accuracy for slopes as the original WENO scheme cannot be achieved. At the same time, the equilibrium state in space and time in HGKS has to be reconstructed separately. Secondly, it is complicated to get reconstructed data at Gaussian points from the WENO-type method in high dimensions. For HGKS, besides the point-wise values at the Gaussian points it also requires the slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previous HGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO with adaptive order (WENO-AO) [2] method is implemented for reconstruction. Equipped with WENO-AO reconstruction, the performance enhancement of HGKS is fully explored. WENO-AO not only provides the interface values, but also the slopes. In other words, a whole polynomial inside each cell is provided by the WENO-AO reconstruction. The available polynomial may not benefit to the high-order schemes based on the Riemann solver, where only points-wise values at the cell interface are needed. But, it can be fully utilized in the HGKS. As a result, the HGKS becomes simpler than the previous one with the direct implementation of cell interface values and their slopes from WENO-AO. The additional reconstruction of equilibrium state at the beginning of each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes the above existing problems in the previous HGKS. The accuracy of the scheme from 1D to 3D from the new HGKS can recover the theoretical order of accuracy of the WENO reconstruction. In the two- and three-dimensional simulations, the new HGKS shows better robustness and efficiency than the previous scheme in all test cases.     

High Order Discretely Well-Balanced Methods for Arbitrary Hydrostatic Atmospheres

We introduce novel high order well-balanced finite volume methods for thefull compressible Euler system with gravity source term. They require no à prioriknowledge of the hydrostatic solution which is to be well-balanced and are not restricted to certain classes of hydrostatic solutions. In one spatial dimension we construct a method that exactly balances a high order discretization of any hydrostaticstate. The method is extended to two spatial dimensions using a local high order approximation of a hydrostatic state in each cell. The proposed simple, flexible, androbust methods are not restricted to a specific equation of state. Numerical tests verifythat the proposed method improves the capability to accurately resolve small perturbations on hydrostatic states.     

Evaluation of Selected Finite-Difference and Finite-Volume Approaches to Rotational Shallow-Water Flow

The shallow-water equations in a rotating frame of reference are importantfor capturing geophysical flows in the ocean. In this paper, we examine and comparetwo traditional finite-difference schemes and two modern finite-volume schemes forsimulating these equations. We evaluate how well they capture the relevant physicsfor problems such as storm surge and drift trajectory modelling, and the schemes areput through a set of six test cases. The results are presented in a systematic mannerthrough several tables, and we compare the qualitative and quantitative performancefrom a cost-benefit perspective. Of the four schemes, one of the traditional finite-difference schemes performs best in cases dominated by geostrophic balance, and oneof the modern finite-volume schemes is superior for capturing gravity-driven motion.The traditional finite-difference schemes are significantly faster computationally thanthe modern finite-volume schemes.     

A Nonlinear Finite Volume Scheme Preserving Maximum Principle for Diffusion Equations

In this paper we propose a new nonlinear cell-centered finite volume scheme on general polygonal meshes for two dimensional anisotropic diffusion problems, which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative.     

Time Implicit Unified Gas Kinetic Scheme for 3D Multi-Group Neutron Transport Simulation

In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D multi-group neutron transport equation with delayed neutron is developed. The explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order Crank-Nicholson scheme, become the subsets of the current IUGKS. In neutron transport, the microscopic angular flux and the macroscopic scalar flux are fully coupled in an implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step. Mathematically, the current scheme has the asymptotic preserving (AP) property in recovering automatically the diffusion solution in the continuum regime. Since the explicit scanning along neutron traveling direction within the computational domain is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other schemes, such as the explicit UGKS, the commonly used finite difference, and finite volume methods. This study shows that the IUGKS can be used faithfully to study neutron transport in practical engineering applications.     

A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep thisbalance exactly due to their unbalanced truncation errors. In this study, we introducean auxiliary variable which becomes constant at isothermal hydrostatic equilibria andpropose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Throughreformulating the convection term and the force term via the auxiliary variable, zeronumerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force.Several problems are tested to demonstrate the accuracy and the stability of the newscheme. The results confirm that, the new scheme can preserve the exact hydrostaticsolution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process ofconverging towards hydrostatic equilibria from a highly unbalanced initial condition.The ultimate state of zero velocity and constant temperature is achieved up to machineaccuracy. As demonstrated by the numerical experiments, the current scheme is verysuitable for small amplitude perturbation and long time running under gravitationalpotential.     

Mesh Adaptation for Curing the Pathological Behaviors of an Upwind Scheme

In present paper, mesh adaptation is applied for curing the pathologicalbehaviors of the enhanced time-accurate upwind scheme (Loh & Jorgenson, AIAAJ2016). In the original ETAU (enhanced time-accurate upwind) scheme, a multi-dimensional dissipation model is required to cure the pathological behaviors. Themulti-dimensional dissipation model will increase the global dissipation level reducing numerical resolution. In present work, the metric-based mesh adaptation strategyprovides an alternative way to cure the pathological behaviors of the shock capturing. The Hessian matrix of flow variables is applied to construct the metric, whichrepresents the curvature of the physical solution. The adapting operation can well refine the anisotropic meshes at the location with large gradients. The numerical resultsshow that the adaptation of mesh provides a possible way to cure the pathologicalbehaviors of upwind schemes.     

A Mixed Finite Element Scheme for Quad-Curl Source and Eigenvalue Problems

The quad-curl problem arises in the resistive magnetohydrodynamics (MHD) and the electromagnetic interior transmission problem. In this paper we study a new mixed finite element scheme using Nédélec's edge elements to approximate both the solution and its curl for quad-curl problem on Lipschitz polyhedral domains. We impose element-wise stabilization instead of stabilization along mesh interfaces. Thus our scheme can be implemented as easy as standardNédélec's methods for Maxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We also extend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numerical examples are provided to show the viability and accuracy of the proposed method for quad-curl source problem.     

High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions

The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general geometries, such as the flow over a wing-body, the development of HGKS in general curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and is transformed back to the physical domain for the update of flow variables inside each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic scheme, the point values as well as the spatial derivatives of conservative variables at Gaussian quadrature points have to be used in the evaluation of the time dependent flux function. The point-wise conservative variables are obtained by ratio of the above reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the accuracy tests to the solutions with strong discontinuities are presented to validate the accuracy and robustness of the current scheme for both inviscid and viscous flows. The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is also demonstrated through the numerical example.     

A WENO-Based Stochastic Galerkin Scheme for Ideal MHD Equations with Random Inputs

In this paper, we investigate the ideal magnetohydrodynamic (MHD) equations with random inputs based on generalized polynomial chaos (gPC) stochasticGalerkin approximation. A special treatment with symmetrization is carried out forthe gPC stochastic Galerkin method so that the resulting deterministic gPC Galerkinsystem is provably symmetric hyperbolic in the spatially one-dimensional case. Wediscretize the hyperbolic gPC Galerkin system with a high-order path-conservative finite volume weighted essentially non-oscillatory scheme in space and a third-order total variation diminishing Runge-Kutta method in time. The method is also extended totwo spatial dimensions via the operator splitting technique. Several numerical examples are provided to illustrate the accuracy and effectiveness of the numerical scheme.     

A Colocalized Scheme for Three-Temperature Grey Diffusion Radiation Hydrodynamics

A positivity-preserving, conservative and entropic numerical scheme is presented for the three-temperature grey diffusion radiation hydrodynamics model. More precisely, the dissipation matrices of the colocalized semi-Lagrangian scheme are defined in order to enforce the entropy production on each species (electron or ion) proportionally to its mass as prescribed in [34]. A reformulation of the model is then considered to enable the derivation of a robust convex combination based scheme. This yields the positivity-preserving property at each sub-iteration of the algorithm while the total energy conservation is reached at convergence. Numerous pure hydrodynamics and radiation hydrodynamics test cases are carried out to assess the accuracy of the method. The question of the stability of the scheme is also addressed. It is observed that the present numerical method is particularly robust.     

An Improved Gas-Kinetic Scheme for Multimaterial Flows

The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstruction method at a cell interface. The iteration method is used to determine the velocity of fluid interface, based on the force balance between both sides due to the incidence and bounce back of particles at the interface. An improved Aitken method is proposed with a simple hybrid of the modified Aitken method (Aitken-Chen) and the Steffensen method. Numerical tests validate its efficiency with significantly less calls to the function not only for the average number but also for the maximum. The new reconstruction is based on the tangent of hyperbola for interface capturing (THINC) but applied only to the volume fraction, which is very simple to be implemented under the stratified framework and capable of resolving fluid interface in mixture. Furthermore, the directional splitting is adopted rather than the previous quasi-one-dimensional method. Typical numerical tests, including several water-gas shock tube flows, and the shock-water cylinder interaction flow show that the improved gas-kinetic scheme can capture fluid interfaces much sharper, while preserving the advantages of the original one.     

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.