## Development and practical implementation of high-order numerical methods for hyperbolic conservation laws

Responsible: V.A. Titarev, Ph.D.

The present research is devoted to the development of accurate numerical methods for solving multi-dimensional problems of rarefied gas dynamics and aerodynamics with the use of unstructured methods and high-performance multi-processor machines. The main results have been published in leading Russian and Western refereed journals and can be grouped into four larger topics as follows.

## 1. Advection schemes for hyperbolic conservation laws

At present the so-called Total Variation Diminishing (TVD) methods, based on Godunov-Kolgan approach, appear to the most popular class of methods for solving numerically conservation equations of hyperbolic type. This type of methods is well suited for flows with strong shock waves, but not very accurate for unsteady long time evolution problems with complicated structures. Examples of such problems include aeroacoustics, turbulence modeling, and seismology. The slope/flux limiters used in TVD method result in clipping of extrema and may lead to unacceptably large errors for such problems.

In recent years there has been a lot of activity in developing uniformly high-order accurate essentially non-oscillatory (ENO) methods. Unlike TVD schemes, such methods do not reduce the order of approximation at extrema and are thus ideally suited for unsteady turbulence modeling. In 2002-2007 a special class of such methods, called ADER (Advection-Diffusion-Reaction), has been extensively developed [1-7], as an extension of the original ADER scheme of Toro et al for the linear constant coefficient equation. ADER methods are based on the approximate solution of the Riemann problem with arbitrary initial conditions. As a result, ADER methods employ one-step simultaneous discretization of convective, diffusive and source terms with arbitrary order of accuracy in time and space. The schemes have been developed for both structured and unstructured 2D/3D meshes. The methods have also been extended to the discontinuous Galerkin (DG) framework.. More recently, Titarev was working on further extension to the arbitrary unstructured meshes of WENO-type methods with Runge-Kutta time marching [8-10]. Such methods are somewhat less efficient than the ADER approach, but simpler and easier to implement.

## 2. MUSTA numerical fluxes

Most popular nowadays Godunov-type upwind methods make heavy use of the Riemann problem solution, either approximate or exact. However, for some equations, such as multiphase models, non-linear elasticity, development and/or program implementation of such solutions is very non-trivial. For such conservation laws MUSTA upwind fluxes offer an attractive alternative. The original idea was suggested by E.F. Toro and resulted in an accurate centred method, with some of the typical shortcomings. Для таких уравнений удобны противопоточные схемы, основанные на многошаговом решателе задачи о распаде разрыва типа MUSTA. In [11-12] the scheme was further improved into an upwind method and a detailed analysis of its properties was carried out. During the development of MUSTA methods an exact solution of the Riemann problem for 2D non-linear elasticity equations was developed, which can be used for verification of numerical methods [13-14]. At present, MUSTA schemes find applications in MHD, multiphase modeling and some other areas.

## 3. Numerical methods in rarefied gas dynamics

The accurate numerical solution of the Boltzmann kinetic equation with the exact or model collision integrals is important in mathematical modeling of gaseous flows inside micro-scale systems as well as for high-altitude flows, typical of re-entry. Past few years have seen rapid development of numerical methods and associated computer codes for solving for three-dimensional problems in the work of Aristov, Zabelok,Frolova, Kolobov, Tcheremisin, and others. Nonetheless, none of these methods fulfill of the requirements for the fast and economical solution of real-world problems with complex geometries.

In 2007-2011 Titarev developed and implemented a new framework for obtaining three-dimensional solutions of the kinetic equation with model collision integrals [15-20], such as models of Shakhov and Rykov.

The framework consists of the three main blocks: high-order accurate implicit advection scheme on hybrid unstructured meshes, conservative procedure for the calculation of the model collision integral and a simple and efficient implementation on modern high-performance clusters. The use of unstructured meshes in physical space simplifies the computations in three-dimensional domains of complex geometry and allows for efficient and accurate resolution of near-wall layers. The high-order accurate total variation diminishing (TVD) advection scheme works well for both large Knudsen numbers, when discontinuities of distribution function play an important role, and for moderate and small Knudsen numbers, for which the high order of accuracy is important. The one-step implicit time discretization method accelerates convergence to a steady state by at least an order of magnitude as compared with explicit time evolution methods. Finally, good scalability of the method makes it possible to use relatively fine meshes with moderate computational time required, which was tested on the HPC systems of Lomonosov Moscow State University and Moscow Institute of Physics and Technology.

## 4. Practical implementation of the methods

Most of the above-mentioned methods have been implemented in the software package "Nesvetay", which allows to solve three-dimensional problems of aerodynamics and rarefied gas dynamics on modern HPC systems.

The package consists of "Nesvetay-2D" and "Nesvetay-3D" parts. In particular, "Nesvetay-3D" contains modules to solve compressible Euler equations as well as various model kinetic equations (BGK, Shakhov, Rykov models) on mixed-element unstructured meshes in three space dimensions.

Main features include:

- Use of arbitrary mixed element meshes, including tetrahedral, prismatic, hexahedral and pyramidal shapes.
- Support of Gambit’s “neutral” and StarCD file formats
- A variety of modern Godunov-type methods for advection part, including TVD methods and WENO-type methods
- Support of rotating frame of reference for compressible Euler equations
- One-step implicit time evolution for model kinetic equations
- OpenMP parallelization for all solvers
- Efficient MPI parallelization in velocity space for the kinetic solver, excellent scalability up to 512 cores
- omain-decomposition type MPI parallelization for all solvers.

At present, the Nesvetay 3D package is further developed for the hypersonic aerodynamics of re-entry vehicles.

### Key publications:

- E.F. Toro and V.A. Titarev. Solution of the generalised Riemann problem for advection-reaction equations. Proc. Roy. Soc. London, 458(2018):271--281, 2002.
- V.A. Titarev and E.F. Toro. ADER: Arbitrary High Order Godunov Approach. J. Sci. Comput., 17:609--618, 2002.
- V.A. Titarev and E.F. Toro. Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys., 201(1):238--260, 2004.
- V.A. Titarev and E.F. Toro. ADER schemes for three-dimensional nonlinear hyperbolic systems., 204(2):715--736, 2005.
- E.F. Toro and V.A. Titarev. Derivative Riemann solvers for systems of conservation laws and ADER methods. J. Comput. Phys., 212(1):150--165, 2006.
- M. Dumbser, M. Kaser, V.A. Titarev, and E.F. Toro. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys., 226:204--243, 2007.
- G. Vignoli, E.F. Toro, and V.Titarev. ADER schemes for the shallow water equations in channel with irregular bottom elevation. J. Comput. Phys. 227:2463--2480, 2008.
- V.A. Titarev, P. Tsoutsanis, and D. Drikakis. WENO schemes for mixed-element unstructured meshes. Communications in Computational Physics, 8(3):585--609, 2010.
- P. Tsoutsanis, V.A. Titarev, and D. Drikakis. WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions. J. Comput. Phys., 230:1585 -- 1601, 2011.
- V.A. Titarev and D.Drikakis. Uniformly high-order schemes on arbitrary unstructured meshes for advection-diffusion equations. Computers and Fluids, 46(1):467--471, 2011. 10th ICFD Conference Series on Numerical Methods for Fluid Dynamics (ICFD 2010).
- V.A. Titarev and E.F. Toro. MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements. International Journal for Numerical Methods in Fluids, 49(2):117--147, 2005.
- E.F. Toro and V.A. Titarev. MUSTA schemes for systems of conservation laws. J. Comput. Phys., 216(2):403--429, 2006.
- V.A. Titarev, E.I. Romenski, and E.F. Toro. MUSTA-type upwind fluxes for non-inear elasticity. Int. J. Num. Methods in Eng., 73:897--926, 2008.
- P.T. Barton, D. Drikakis, E. Romenski, and V.A. Titarev. Exact and approximate solutions of Riemann problems in non-linear elasticity. J. Comput. Phys., 228:7046 -- 7068, 2009.
- V.A. Titarev Conservative numerical methods for model kinetic equations. Computers and Fluids, 36(9):1446 -- 1459, 2007.
- V.A. Titarev. Numerical method for computing two-dimensional unsteady rarefied gas flows in arbitrarily shaped domains. Computational Mathematics and Mathematical Physics Vol. 49, No. 7, pp. 1197–1211, 2009
- V.A. Titarev. Implicit unstructured-mesh method for calculating Poiseuille flows of rarefied gas. Communications in Computational Physics, 8(2):427--444, 2010.
- V.A. Titarev Implicit numerical method for computing three-dimensional rarefied gas flows using unstructured meshes. Computational Mathematics and Mathematical Physics. V. 50, N. 10, pp. 1719–1733, 2010.
- V.A. Titarev. Rarefied flow in a long planar microchannel of finite length. J. Comput. Phys., 231(1):109--134, 2012.
- V.A. Titarev. Efficient deterministic modelling of three-dimensional rarefied gas flows. Communications in Computational Physics, V. 12, N. 1, p. 161-192, 2012.