Extension and Comparative Study of AUSM-Family Schemes for Compressible Multiphase Flow Simulations

Several recently developed AUSM-family numerical flux functions (SLAU, SLAU2, AUSM⁺-up2, and AUSMPW+) have been successfully extended to compute compressible multiphase flows, based on the stratified flow model concept, by following two previous works: one by M.-S. Liou, C.-H. Chang, L. Nguyen, and T.G. Theofanous [AIAA J. 46:2345-2356, 2008], in which AUSM⁺-up was used entirely, and the other by C.-H. Chang, and M.-S. Liou [J. Comput. Phys. 225:840-873, 2007], in which the exact Riemann solver was combined into AUSM⁺-up at the phase interface. Through an extensive survey by comparing flux functions, the following are found: (1) AUSM⁺-up with dissipation parameters of K_p and K_u equal to 0.5 or greater, AUSMPW+, SLAU2, AUSM⁺-up2, and SLAU can be used to solve benchmark problems, including a shock/water-droplet interaction; (2) SLAU shows oscillatory behaviors [though not as catastrophic as those of AUSM⁺ (a special case of AUSM⁺-up withK_p=K_u=0)] due to insufficient dissipation arising from its ideal-gas-based dissipation term; and (3) when combined with the exact Riemann solver, AUSM⁺-up (K_p=K_u=1), SLAU2, and AUSMPW+ are applicable to more challenging problems with high pressure ratios.     

A High-Order Accurate Gas-Kinetic Scheme for One- and Two-Dimensional Flow Simulation

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the "initial" cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.     

A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations

In this paper, a compact third-order gas-kinetic scheme is proposed for the compressible Euler and Navier-Stokes equations. The main reason for the feasibility to develop such a high-order scheme with compact stencil, which involves only neighboring cells, is due to the use of a high-order gas evolution model. Besides the evaluation of the time-dependent flux function across a cell interface, the high-order gas evolution model also provides an accurate time-dependent solution of the flow variables at a cell interface. Therefore, the current scheme not only updates the cell averaged conservative flow variables inside each control volume, but also tracks the flow variables at the cell interface at the next time level. As a result, with both cell averaged and cell interface values, the high-order reconstruction in the current scheme can be done compactly. Different from using a weak formulation for high-order accuracy in the Discontinuous Galerkin method, the current scheme is based on the strong solution, where the flow evolution starting from a piecewise discontinuous high-order initial data is precisely followed. The cell interface time-dependent flow variables can be used for the initial data reconstruction at the beginning of next time step. Even with compact stencil, the current scheme has third-order accuracy in the smooth flow regions, and has favorable shock capturing property in the discontinuous regions. It can be faithfully used from the incompressible limit to the hypersonic flow computations, and many test cases are used to validate the current scheme. In comparison with many other high-order schemes, the current method avoids the use of Gaussian points for the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping technique. Due to its multidimensional property of including both derivatives of flow variables in the normal and tangential directions of a cell interface, the viscous flow solution, especially those with vortex structure, can be accurately captured. With the same stencil of a second order scheme, numerical tests demonstrate that the current scheme is as robust as well-developed second-order shock capturing schemes, but provides more accurate numerical solutions than the second order counterparts.     

Improved rat liver preservation by hypothermic continuous machine perfusion using polysol, a new, enriched preservation solution.

For experimental machine perfusion (MP) of the liver, the modified University of Wisconsin solution (UW-G) is most often used. In our search for an enriched MP preservation solution, Polysol was developed. Polysol is enriched with various amino acids, vitamins, and other nutrients for the liver metabolism. The aim of this study was to compare Polysol with UW-G for MP preservation of the liver. Rat livers were preserved during 24 hours with hypothermic MP using UW-G (n = 5) or Polysol (n = 5). Hepatocellular damage (aspartate aminotransferase [AST], alanine aminotransferase [ALT], lactate dehydrogenase [LDH], alpha-glutathione-S-transferase [alpha-GST]) and bile production were measured during 60 minutes of reperfusion (37 degrees C) with Krebs-Henseleit buffer. Control livers were reperfused after 24 hours of cold storage in UW (n = 5). MP using UW-G or Polysol showed less liver damage when compared with controls. Livers machine perfused with Polysol showed less enzyme release when compared to UW-G. Bile production was higher after MP using either UW-G or Polysol compared with controls. In conclusion, machine perfusion using Polysol results in better quality liver preservation than cold storage with UW and machine perfusion using UW-G.     

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.     

High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

In this paper, the second-order and third-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO limiter is an extension of a finite volume multi-resolution WENO scheme developed in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and third-order RKDG methods with the associated multi-resolution WENO limiters are developed as examples for general high-order RKDG methods, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong discontinuities by gradually degrading from the optimal order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be set as any positive numbers on the condition that they sum to one. A series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on tetrahedral meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes) and three-dimensional tests are performed to demonstrate the good performance of the RKDG methods with new multi-resolution WENO limiters.     

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

In this paper, a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J. Comput. Phys., 446 (2021) 110653], in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries. However, in this paper, only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments. In addition, an extra modification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolution near discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights are not unique and are independent of the node position, and the CFL number can still be 0.6 whether for the one or two dimensional case, which has to be 0.2 in the two dimensional case for other HWENO schemes. Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.     

A Multi-Domain Hybrid DG and WENO Method for Hyperbolic Conservation Laws on Hybrid Meshes

In [SIAM J. Sci. Comput., 35(2)(2013), A1049–A1072], a class of multi-domain hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux with Lagrangian interpolation is applied as the interface flux during the moment of conservative coupling. In this continuation paper, we present a more robust approach in the construction of DG flux at the coupling interface by using WENO procedures of reconstruction. Based on this approach, such numerical instability is overcome very well. In addition, the procedure of coupling a DG method with a WENO-FD scheme on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either WENO-FD flux or DG flux at the moment of requiring conservative coupling.     

Subnormothermic machine perfusion protects against rat liver preservation injury: a comparative evaluation with conventional cold storage

Hypothermic machine perfusion (MP) of the liver has been reported to improve graft function reclaiming marginal livers, such as those from non-heart-beating donors. Livers from obese donors often have fatty infiltrates and are more susceptible to hypothermic conditions. No data exist about MP at temperatures >4 degrees C. This study evaluated liver function after organ preservation by comparing MP at 20 degrees C with conventional cold storage. METHODS: For MP, rat livers were perfused for 6 hours using an oxygenated Krebs-Henseleit (KH) solution at 20 degrees C (pH 7.4). For cold storage, livers were perfused in situ and preserved with Celsior solution at 4 degrees C for 6 hours. The reperfusion period with KH (2 hours at 37 degrees C) was performed under the same conditions both among livers preserved by MP or cold storage. Hepatic enzyme release (aspartate aminotransferase [AST], alanine aminotransferase [ALT], lactate dehydrogenase [LDH], and gamma-glutamyl transferase [GGT]), bile production, and ATP levels were measured during MP and reperfusion. RESULTS: At the end of reperfusion, livers preserved by MP showed significantly decreased liver damage compared with cold storage: AST, 18 +/- 4 vs. 45 +/- 6 mU/mL (P < .01); ALT, 1.5 +/- .07 vs. 6 +/- 0.5 mU/mL (P < .01); and LDH, 82 +/- 2 vs. 135 +/- 29 mU/mL (P < .05). No difference was observed between bile production between MP and cold storage. High levels of biliary GGT and LDH were found in cold preserved livers. ATP levels were higher in livers preserved with MP compared with those preserved by cold storage. CONCLUSIONS: MP at 20 degrees C resulted in a better quality of liver preservation, improving hepatocyte survival, compared with conventional cold storage. This may provide a new method for successful utilization of marginal livers, in particular fatty livers.     

The influence of pulsatile preservation on renal transplantation in the 1990s

BACKGROUND: Unlike simple cold storage (CS), pulsatile machine preservation (MP) of kidneys for transplantation permits pharmacologic manipulation of the perfusate and aids in the pretransplant assessment of the kidney graft. These characteristics of MP may have importance in the era of increasing use of extended criteria donor kidneys. The overall aim of this article is to critically assess practices at our preservation unit with respect to graft function. Specific aims are to (1) compare the influence of MP versus CS on graft function, (2) determine which pretransplant variables have significance in pretransplant assessment, and (3) determine whether pharmacologic manipulation during MP is advantageous. METHODS: There were 650 consecutive kidneys preserved in our laboratory between January 1, 1993 and March 1, 999, by either MP or CS. All MP kidneys were preserved by continuous hypothermic pulsatile perfusion using Belzer-MPS or Belzer II solution. Perfusion parameters and electrolytes were measured serially during pulsatile perfusion. All CS kidneys were stored in University of Wisconsin solution. All kidneys obtained from donors exhibiting extended criteria features underwent pretransplant frozen section biopsies. Transmission electron microscopy (EM) was performed on a subset of kidneys undergoing pharmacologic manipulation. Four agents were assessed prospectively for their ability to influence MP characteristics when added to perfusate: PGE1, trifluoperazine, verapamil, and papaverine. RESULTS: MP was associated with improved immediate, 1-, and 2-year graft function and reduced length of initial hospital stay when compared with CS grafts. Changes in the machine perfusion variables flow and resistance, and the [Ca++] in perfusate, were significantly associated with delayed graft function (DGF) after the transplant. Biopsy information was not predictive of DGF. The addition of PGE1 to perfusate improved MP characteristics, reduced the release of [Ca++] into perfusate, and ameliorated mitochondrial ischemic injury in transmission EM images. Early graft function was improved in the presence of PGE1+MP, compared with function in the presence of other pharmacologic agents or CS alone. CONCLUSIONS: MP is associated with improved early and long term renal function. Moreover, PGE1 augments MP in improving graft function. The combination of MP+PGE1 may be important in optimizing the ability to use extended donor criteria kidneys and, thereby, improve the overall efficiency of cadaveric renal transplantation.     

A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations

Based on the high order essentially non-oscillatory (ENO) Lagrangian type scheme on quadrilateral meshes presented in our earlier work [3], in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics. The main purpose of this work is to demonstrate our claim in [3] that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges, which restricts the accuracy of the resulting scheme to at most second order. The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes. Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties.     

Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes

In this article we present two types of nonlinear positivity-preserving finite volume (PPFV) schemes for a class of three-dimensional heat conduction equations on general polyhedral meshes. First, we present a new parameter selection strategy on the one-sided flux and establish a nonlinear PPFV scheme based on a two-point flux with higher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai, H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes our interpolation formulae suitable to be constructed by existing approaches with high accuracy and well robustness (e.g., the finite element method), thus enhancing the adaptability to distorted meshes with large deformations. Then we derive a linear multi-point flux involving combination coefficients and, via the Patankar trick, obtain another nonlinear PPFV scheme that is concise and easy to implement. The selection strategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preserving properties of these two nonlinear PPFV solutions are proved. Numerical experiments with the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that our schemes both can achieve ideal-order accuracy even on severely distorted meshes.     

Extension of the High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme to Solve Time Dependent Diffusion Equations

In this paper, the high-order space-time discontinuous Galerkin cell vertex scheme (DG-CVS) developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion equations. In the extension, the treatment of the diffusive flux is exactly the same as that for the advective flux. Thanks to the Riemann-solver-free and reconstruction-free features of DG-CVS, both the advective flux and the diffusive flux are evaluated using continuous information across the cell interface. As a result, the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common "variational crime" problem when traditional DG methods are applied to diffusion problems. For this reason, DG-CVS is conceptually simpler than other existing DG-typed methods. The numerical tests demonstrate that the convergence order based on the L²-norm is optimal, i.e. O(h^p+1) for the solution and O(h^p) for the solution gradients, when the basis polynomials are of odd degrees. For even-degree polynomials, the convergence order is sub-optimal for the solution and optimal for the solution gradients. The same odd-even behaviour can also be seen in some other DG-typed methods.     

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.     

Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes

In this paper, we introduce a high-order accurate constrained transport type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional triangular meshes. A new divergence-free WENO-based reconstruction method is developed to maintain exactly divergence-free evolution of the numerical magnetic field. In this formulation, the normal component of the magnetic field at each face of a triangle is reconstructed uniquely and with the desired order of accuracy. Additionally, a new weighted flux interpolation approach is also developed to compute the z-component of the electric field at vertices of grid cells. We also present numerical examples to demonstrate the accuracy and robustness of the proposed scheme.     

Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations

This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is O(h⁴) for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an O(h⁴+k⁴) accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one- and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.     

On the Spurious Mode Generation Induced by Spectral-Like Optimized Interpolation Schemes Used in Computational Acoustics

The present work constitutes a fraction of a more extensive study that is devoted to numerical methods in acoustics. More precisely, we address here the interpolation process, which is more and more frequently used in Computational Acoustics – whether it is for enabling multi-stage hybrid calculations, or for easing the proper handling of complex configurations via advanced techniques such as Chimera grids or Immersed Boundary Conditions. In that regard, we focus on high-order interpolation schemes, so as to analyze their intrinsic features and to assess their effective accuracy. Taking advantage of specific insights that had been previously achieved by the present authors regarding standard high-order interpolation schemes (of centered nature), we here focus on their so-called spectral-like optimized counterparts (of both centered and noncentered nature). The latter spectral-like optimized schemes are analyzed thoroughly thanks to dedicated theoretical developments, which allow highlighting better what their strengths and weaknesses are. Among others, the various ways such interpolation schemes can degrade acoustic signals they are applied to are carefully investigated from a theoretical point-of-view. Besides that, specific criteria that could help in optimizing interpolation schemes better are provided, along with generic rules about how to minimize the signal degradation induced by existing interpolation schemes, in practice.     

A Compact High Order Space-Time Method for Conservation Laws

This paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.     

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 moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. 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 or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. 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 treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for 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 and its ability to adapt the mesh according to features in the flow and bottom topography.     

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 and provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation on rectangular meshes, the extension to triangular meshes is conceptually plausible but highly nontrivial. We first introduce a special way to recover the equilibrium state and then design a group of novel variables at the interface of two adjacent cells, which plays an important role in the well-balanced and positivity-preserving properties. One main challenge is that the well-balanced schemes may not have the weak positivity property. In order to achieve the well-balanced and positivity-preserving properties simultaneously 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 positivity property. A simple existing limiter can be applied to enforce the positivity-preserving property, without losing high-order accuracy and conservation. Extensive one- and two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness.