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 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.     

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 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.     

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.     

Linear Stability Analysis and Gas Kinetic Scheme (GKS) Simulations of Instabilities in Compressible Plane Poiseuille Flow

The fundamental nature of flow instability in wall bounded flows changeswith Mach number. The objectives of this study are two-fold, (i) compute the instability modes in high Mach number Poiseuille flows using linear stability analysis (LSA)and, (ii) perform direct numerical simulations (DNS) of the instability developmentusing a solver based on gas kinetic method (GKM) for the purpose of code validationby comparison against LSA results. The LSA and DNS are performed for the case ofPoiseuille flow over a range of Mach numbers – from moderately supersonic to hypersonic speeds. First, LSA is employed to identify the most unstable mode over the rangeof Mach numbers. We then perform two sets of GKM-DNS to corroborate the LSA results over the Mach number range. In the first set of simulations, the background fieldis initially perturbed with the most unstable mode identified by LSA and the evolutionis monitored. It is shown that GKM-DNS accurately captures the exponential growthin kinetic energy for all Mach numbers. The second set of GKM-DNS simulations isperformed by superposing the background pressure field with random initial perturbations. After an initial transient period, the modes predicted by LSA dominate theDNS flow field evolution. The wave-vector and mode shapes of the dominant instability are well replicated by GKM-DNS at each Mach number. These insights in thelinear regime of high speed Poiseuille flow and validation of GKM are important forunderstanding and simulating wall bounded flows.     

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper, we propose a high-order accurate discontinuous Galerkin(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state andprovably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitationon rectangular meshes, the extension to triangular meshes is conceptually plausiblebut highly nontrivial. We first introduce a special way to recover the equilibrium stateand then design a group of novel variables at the interface of two adjacent cells, whichplays an important role in the well-balanced and positivity-preserving properties. Onemain challenge is that the well-balanced schemes may not have the weak positivityproperty. In order to achieve the well-balanced and positivity-preserving propertiessimultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivityproperty. A simple existing limiter can be applied to enforce the positivity-preservingproperty, without losing high-order accuracy and conservation. Extensive one- andtwo-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness.     

A Hybrid WENO Scheme for Steady Euler Equations in Curved Geometries on Cartesian Grids

For steady Euler equations in complex boundary domains, high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian grids to achieve uniform high-order accuracy. In this paper, we utilize a fifth-order finite difference hybrid WENO scheme to simulate steady Euler equations, and the same fifth-order WENO extrapolation methods are developed to handle the curved boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform the tangential derivatives along the curved solid wall boundary. This hybrid WENO scheme is robust for steady-state convergence and maintains high-order accuracy in the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows that this hybrid WENO scheme has low dispersion and dissipation errors. Numerical examples are presented to validate the high-order accuracy and robust performance of the hybrid scheme for steady Euler equations in curved domains with Cartesian grids.     

A New Interpolation for Auxiliary Unknowns of the Monotone Finite Volume Scheme for 3D Diffusion Equations

A monotone cell-centered finite volume scheme for diffusion equations ontetrahedral meshes is established in this paper, which deals with tensor diffusion coefficients and strong discontinuous diffusion coefficients. The first novelty here is topropose a new method of interpolating vertex unknowns (auxiliary unknowns) withcell-centered unknowns (primary unknowns), in which a sufficient condition is givento guarantee the non-negativity of vertex unknowns. The second novelty of this paperis to devise a modified Anderson acceleration, which is based on an iterative combination of vertex unknowns and will be denoted as AA-Vertex algorithm, in order to solvethe nonlinear scheme efficiently. Numerical testes indicate that our new method canobtain almost second order accuracy and is more accurate than some existing methods.Furthermore, with the same accuracy, the modified Anderson acceleration is muchmore efficient than the usual one.     

Vectorial Kinetic Relaxation Model with Central Velocity. Application to Implicit Relaxations Schemes

We apply flux vector splitting (FVS) strategy to the implicit kinetic schemesfor hyperbolic systems. It enables to increase the accuracy of the method compared toclassical kinetic schemes while still using large time steps compared to the characteristic speeds of the problem. The method also allows to tackle multi-scale problems, suchas the low Mach number limit, for which wave speeds with large ratio are involved. Wepresent several possible kinetic relaxation schemes based on FVS and compare themon one-dimensional test-cases. We discuss stability issues for this kind of method.     

Upwind Biased Local RBF Scheme with PDE Centres for the Steady Convection Diffusion Equations with Continuous and Discontinuous Boundary Conditions

RBF based grid-free scheme with PDE centres is experimented in this workfor solving Convection-Diffusion Equations (CDE), a simplified model of the Navier-Stokes equations. For convection dominated problems, very few integration schemescan give converged solutions for the entire range of diffusivity wherein sharp layers areexpected in the solutions and accurate computation of these layers is a big challengefor most of the numerical schemes. Radial Basis Function (RBF) based Local HermitianInterpolation (LHI) with PDE centres is one such integration scheme which has somebuilt in upwind effect and hence may be a good solver for the convection dominatedproblems. In the present work, to get convergent solutions consistently for small diffusion parameters, an explicit upwinding is also introduced in to the RBF based schemewith PDE centres, which was initially used to solve some time dependent problemsin [10]. RBF based numerical schemes are one type of grid free numerical schemesbased on the radial distances and hence very easy to use in high dimensional problems. In this work, the RBF scheme, with different upwind biasing, is used to a varietyof steady benchmark problems with continuous and discontinuous boundary data andvalidated against the corresponding exact solutions. Comparisons of the solutions ofthe convective dominant benchmark problems show that the upwind biasing eitherin source centres or PDE centres gives convergent solutions consistently and is stablewithout any oscillations especially for problems with discontinuities in the boundaryconditions. It is observed that the accuracy of the solutions is better than the solutionsof other standard integration schemes particularly when the computations are carriedout with fewer centers.     

Multiple-Temperature Gas-Kinetic Scheme for Type IV Shock/Shock Interaction

In this paper, a gas-kinetic scheme (GKS) method coupled with a three temperature kinetic model is presented and applied in numerical study of the Edney-type IV shock/shock interaction which could cause serious problems in hypersonic vehicles. The results showed very good agreement with the experimental data in predicting the heat flux on the surface. It could be obviously seen that the current method can accurately describe the position and features of supersonic jets structure and clearly capture the thermal non-equilibrium in this case. The three temperature kinetic model includes three different models of temperatures which are translational, rotational and vibrational temperatures. The thermal non-equilibrium model is used to better simulate the aerodynamic and thermodynamic phenomenon. Current results were compared with the experimental data, computational fluid dynamics (CFD) results, and the Direct Simulation Monte Carlo (DSMC) results. Other CFD methods include the original GKS method without considering thermal non-equilibrium, the GKS method with a two temperature kinetic model and the Navier-Stokes equations with a three temperature kinetic model, which is the same as the multiple temperature kinetic model in current GKS method. Comparisons were made for the surface heat flux, pressure loads, Mach number contours and flow field values, including rotational temperature and density. By Comparing with other CFD method, the current GKS method showed a lot of improvement in predicting the magnitude and position of heat flux peak on the surface. This demonstrated the good potential of the current GKS method in solving thermodynamic non-equilibrium problems in hypersonic flows. The good performance of predicting the heat flux could bring a lot of benefit for the designing of the thermal protection system (TPS) for the hypersonic vehicles. By comparing with the original GKS method and the two temperature kinetic model, the three temperature kinetic model showed its importance and accuracy in this case.     

An All Speed Second Order IMEX Relaxation Scheme for the Euler Equations

We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension.     

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.     

Solution Remapping Method with Lower Bound Preservation for Navier-Stokes Equations in Aerodynamic Shape Optimization

It is found that the solution remapping technique proposed in [Numer. Math.Theor. Meth. Appl., 2020, 13(4)] and [J. Sci. Comput., 2021, 87(3): 1-26] does not workout for the Navier-Stokes equations with a high Reynolds number. The shape deformations usually reach several boundary layer mesh sizes for viscous flow, which farexceed one-layer mesh that the original method can tolerate. The direct application toNavier-Stokes equations can result in the unphysical pressures in remapped solutions,even though the conservative variables are within the reasonable range. In this work,a new solution remapping technique with lower bound preservation is proposed toconstruct initial values for the new shapes, and the global minimum density and pressure of the current shape which serve as lower bounds of the corresponding variablesare used to constrain the remapped solutions. The solution distribution provided bythe present method is proven to be acceptable as an initial value for the new shape.Several numerical experiments show that the present technique can substantially accelerate the flow convergence for large deformation problems with 70%-80% CPU timereduction in the viscous airfoil drag minimization.     

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.     

A Gas-Kinetic Unified Algorithm for Non-Equilibrium Polyatomic Gas Flows Covering Various Flow Regimes

In this paper, a gas-kinetic unified algorithm (GKUA) is developed to investigate the non-equilibrium polyatomic gas flows covering various regimes. Basedon the ellipsoidal statistical model with rotational energy excitation, the computablemodelling equation is presented by unifying expressions on the molecular collision relaxing parameter and the local equilibrium distribution function. By constructing thecorresponding conservative discrete velocity ordinate method for this model, the conservative properties during the collision procedure are preserved at the discrete levelby the numerical method, decreasing the computational storage and time. Explicitand implicit lower-upper symmetric Gauss-Seidel schemes are constructed to solvethe discrete hyperbolic conservation equations directly. Applying the new GKUA,some numerical examples are simulated, including the Sod Riemann problem, homogeneous flow rotational relaxation, normal shock structure, Fourier and Couette flows,supersonic flows past a circular cylinder, and hypersonic flow around a plate placednormally. The results obtained by the analytic, experimental, direct simulation MonteCarlo method, and other measurements in references are compared with the GKUAresults, which are in good agreement, demonstrating the high accuracy of the presentalgorithm. Especially, some polyatomic gas non-equilibrium phenomena are observedand analysed by solving the Boltzmann-type velocity distribution function equationcovering various flow regimes.     

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.     

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.     

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressibleNavier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite elementmethod for the space discretization. This class of computational solvers benefits fromthe geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows usingtime steps larger than its Eulerian counterparts. In the current study, we implementthree-dimensional limiters to convert the proposed solver to a fully mass-conservativeand essentially monotonicity-preserving method in addition of a low computationalcost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. Theproposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical resultsillustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominatedflow problems on unstructured tetrahedral meshes.