Elsevier

Wave Motion

Volume 98, November 2020, 102626
Wave Motion

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

https://doi.org/10.1016/j.wavemoti.2020.102626Get rights and content

Abstract

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.

Introduction

Solutions of the nonlinear hyperbolic conservation laws are often discontinuous in nature even when the initial condition is sufficiently smooth. Thus, one must seek the solutions in a weak or distributional sense by abandoning the notion of classical solution. If a classical solution exists, it coincides with the weak solution. Over the past few decades, various numerical methods have been developed for hyperbolic conservation laws. Early attempts mainly focused on high-resolution total variation diminishing shock capturing methods [1]. High-order of accuracy is a desirable property in the solution of partial differential equations. Various methods are proposed to achieve higher order accuracy. Recently, new methods like spectral difference methods [2], discontinuous Galerkin methods [3], [4], [5], spectral element methods [6], [7], [8], [9], finite difference and finite volume based weighted essentially non-oscillatory (WENO) methods [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], higher order shock fitting/capturing methods [20] have been developed, which not only improves the resolution of a smooth region by minimizing the numerical diffusion but also captures discontinuous features like shock waves accurately. In the literature, WENO schemes and their variations like hybrid WENO [21], [22], [23], [24], [25], central WENO or CWENO [26], [27], [28] are extensively used to solve hyperbolic conservation laws. Capability of WENO schemes to achieve arbitrarily high-order accuracy in smooth regions while maintaining non-oscillatory, stable and accurate transitions across discontinuities like shocks and contact waves is one of the main advantages. Such schemes are quite suitable for hyperbolic conservation laws where both strong discontinuity and smooth regions are present in the solution [29], [30], [31], [32]. ENO schemes were first proposed by Harten et al. in [10] whereas Liu et al. first proposed third order finite volume WENO scheme [12]. Later, Jiang and Shu proposed a general framework to construct any arbitrary order accurate finite difference WENO scheme referred to as WENO-JS scheme [11]. As noted by Henrick et al. in [33], the classical weighted procedure used in WENO-JS does not give the required order of convergence at critical points. Moreover, they also obtained the necessary and sufficient conditions on the nonlinear weights due to which WENO-JS does not attain required convergence rate. In their analysis, they achieved optimal order of convergence using the mapping function. This improved scheme is known as mapped WENO or WENO-M scheme. In [34], Borges et al. investigated an improved weighting procedure which resulted in WENO-Z scheme, which is shown to be more efficient than the WENO-M scheme. Castro et al. [35], extended the WENO-Z scheme by generalizing the formula for the higher order smoothness indicators to all odd orders of accuracy. Various other approaches have been suggested to improve the WENO scheme by designing the smoothness indicators. Ha et al. [36], suggested an L1-norm based smoothness indicator, which is employed recently in [37] for nonlinear degenerate parabolic equations. The Lagrangian interpolation polynomial based local smoothness indicator is proposed by Fan et al. in [38], [39]. Adaptive order WENO scheme or WENO-AO scheme were recently proposed by Zhu and Qiu [40] and later improved by Balasara et al. [41]. It is a convex combination of fifth-order linear reconstruction and three third-order linear reconstructions in smooth regions denoted by WENO-AO(5,3) in order to achieve optimal accuracy. Here, smoothness indicators are computed by expressing reconstruction polynomials in terms of Legendre polynomials. These adaptive scheme were shown to be more accurate than previously mentioned WENO schemes, but due to calculation of additional smoothness indicators over the larger stencil, these schemes become computationally expensive. In [42], Kumar & Chandrasekhar proposed WENO-AO(5,4,3) scheme by adding an extra stencil, which is shown to be more accurate than WENO-AO(5,3) scheme. WENO-AO(5,4,3) scheme uses a convex combination of three quadratic polynomials along with one cubic and one quartic polynomial. When a discontinuity is detected over the bigger stencil, the solution is computed using cubic polynomial rather than quadratic polynomial as in the case of WENO-AO(5,3) scheme, which results in a more accurate solution near discontinuities. For more details on adaptive order WENO schemes, see [41], [43], [44]. In WENO schemes, the nonlinear weights are designed to approach linear weights with the requirement that the sum of linear weights becomes unity. One-dimensional WENO formulation is straightforward but, extension of WENO scheme to higher dimensions can be done either by genuine multidimensional reconstruction or by dimension-by-dimension reconstruction. In the first approach, all space coordinates are considered simultaneously at all grid points of the multidimensional stencil to construct numerical flux while in the second approach, one uses one-dimensional sweeps in the reconstruction of numerical flux.

Kinetic schemes (also called as Boltzmann schemes) are interesting numerical methods used to solve hyperbolic conservation laws. Development of these schemes is based on the fact that one can recover the Euler equations by applying a suitable moment method strategy to the scalar Boltzmann equation. Thus, instead of dealing with a nonlinear hyperbolic conservation laws (which are the Euler equations), we are dealing with a simple linear scalar equation hence, the numerical scheme developed for the linear equation can be mapped to a numerical scheme for nonlinear equation like Euler equations. There are many kinetic schemes available in the literature like, Beam scheme of Sanders & Prendergast [45], the method of Rietz [46], the equilibrium flux method of Pullin [47], kinetic flux vector splitting method of Deshpande [48], [49], which is very similar to equilibrium flux method, the compactly supported distribution based methods of Kaniel [50] and Perthame [51], the peculiar velocity based upwind method of Deshpande and co-workers [52], [53] and the BGK scheme of Prendergast & Xu [54]. These methods were developed in the framework of finite difference or finite volume methods. Kinetic theory based splitting technique is also used in WENO framework by many authors like Kumar et al. [55], where they used WENO-enhanced gas-kinetic scheme for simulation of compressible transition and turbulence. Xuan and Xu proposed gas-kinetic scheme based on analytical solutions of BGK equation [56]. In [57], Liu proposed hybrid kinetic WENO scheme for inviscid as well as viscous flows. Recently, Ji et al. proposed fourth order gas-kinetic scheme for Euler and Navier–Stokes equations using Hermite WENO framework [58]. In [59], Xuan and Xu proposed WENO based finite difference gas-kinetic scheme for Euler as well as Navier–Stokes equations.

In this paper, the kinetic theory based splitting schemes are used in various WENO schemes. It can be seen that the accuracy of the WENO schemes is further improved by kinetic theory based flux splitting methods. Some of the salient features of the proposed formulation are

  • 1.

    The kinetic flux vector splitting method is used in adaptive order finite difference WENO framework. In this method, upwinding is done using molecular velocity (v). The molecular velocity space is split into two parts viz., v>0 and v<0. Such splitting involves computationally expensive error and exponential functions.

  • 2.

    To reduce the computational cost, the peculiar velocity based splitting is given, which removes these computationally expensive functions from the splitting.

  • 3.

    In the first part of the numerical experiments, the one-dimensional test cases of compressible Euler equations are solved, where kinetic theory based fluxes are used in various WENO schemes like WENO-JS, WENO-Z, WENO-AO(5,3) and WENO-AO(5,4,3) to obtained the best WENO scheme, and then, using this best WENO scheme, various test cases are solved in one and two dimensions with kinetic theory based flux splitting methods and Lax–Friedrich based splitting, where accuracy of the kinetic based splitting scheme is clearly visible.

  • 4.

    Error analysis of the proposed schemes give optimal convergence rate in various norms for both one- and two-dimensional test cases including isentropic vortex test case.

This paper is arranged as follows: After the introduction in Section 1, Section 2 gives governing compressible Euler equations. Section 3 describes the gas-kinetic theory based splittings where two splitting techniques are discussed in depth. Section 4 describes the multi-level adaptive order finite difference WENO scheme in detail. The TVD Runge–Kutta method for temporal discretization is described in Section 5 where various numerical experiments are carried out, which shows the efficacy of the proposed schemes. Small-scaled structures are captured well by the proposed schemes for various test cases such as Rayleigh–Taylor instability, two-dimensional Riemann problems etc. Finally, Section 6 details the conclusions that are drawn from the results.

Section snippets

Compressible Euler equations

The governing compressible Euler equations with source term is given by Ut+i=1DGi(U)xi=S(U),(x,t)ΩTfΩ×(0,Tf]RD×R+,with appropriate initial and boundary conditions. Fluxes Gi(U):RR are functions of conserved variable U:RD×R+R and S(U):RR is the source term. In two dimensions (D=2), U and Gi’s are given as U=ρρu1ρu2ρE,Gi=ρuiδi1p+ρu1uiδi2p+ρu2uipui+ρuiE,i=1,2,where ρ,u1,u2,E,p are mass density, velocity components in x and y directions, total energy and pressure respectively and δij is

Gas-kinetic theory

In gas-kinetic theory, the governing equation is the Boltzmann equation given by ft+vf=dfdtC,(x,t)ΩTfRD×R+,where f and v=(v1,v2,,vD)T are velocity distribution function and molecular velocity respectively. The right-hand side is the collision term and left-hand side consists of an unsteady term and a convection term. The well-known BGK model simplifies the collision term and converts the integro-differential equation to a partial differential equation with a relaxation source term [60].

Finite difference WENO scheme

This section introduces adaptive fifth-order finite difference WENO scheme denoted by WENO-AO(5,4,3) for one-dimensional scalar conservation laws given by ut+g(u)x=0,(x,t)[xl,xr]×(0,Tf]R×R+,u(x,0)=u0(x), with appropriate boundary conditions.

The computational domain is discretized on a uniform mesh denoted by points xj=jΔx,j=0,1,,N. We define cell Ij, cell centre xj and cell width Δx as follows Ij=[xj12,xj+12],xj=(xj12+xj+12)2,Δx=xj+12xj12.Semi-discretization of Eq. (4.15) over

Numerical experiments

For temporal discretization of the semi-discrete scheme, TVD Runge–Kutta method of order three is used. The objective of the high-order TVD-RK time discretization is to maintain the TVD property while achieving higher order accuracy in time, which turns out to be very useful in numerical schemes such as the method of lines for partial differential equations, especially hyperbolic conservation laws involving discontinuous solution. The semi-discrete scheme for vector conservation laws is written

Conclusions

High-order of accuracy is an essential ingredient for obtaining accurate solution of compressible Euler equations involving discontinuities like shock and contact waves. For this reason, higher order WENO schemes are extensively used to solve these equations. In this paper, the kinetic theory based multi-level adaptive order finite difference WENO schemes namely, WENO-AO-K1 and WENO-AO-K2 are presented. Due to involvement of error and exponential functions in the kinetic flux vector splitting

CRediT authorship contribution statement

Ameya D. Jagtap: Conceptualization, Validation, Supervision, Corrections, Writing - original draft, Writing - review & editing. Rakesh Kumar: Data curation, Validation, Visualization, Supervision, Corrections, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors are grateful to the referees for their careful reading and valuable suggestions. In addition, Rakesh Kumar would like to acknowledge funding support from National Post-Doctoral Fellowship (PDF/2018/002621) administered by SERB-DST, India. The authors also would like to thank Mr. Deepak Varma, TIFR-Centre for Applicable Mathematics, Bengaluru and Ms. Charulatha M., Department of Aerospace Engineering, Indian Institute of Science, Bengaluru for carefully reading the manuscript and

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