Very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control

In this paper, a new family of very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) is proposed. The new framework allows for constructing arbitrarily high-order TENO schemes in a unified paradigm and the yielded nonlinear schemes gradually reduce to low-order reconstructions by judging the smoothness with the ENO-like stencil selection strategy. In order to control the nonlinear numerical dissipation adaptively, the flow scales are first measured by examining the first-order undivided difference and the cut-off constant $C_T$ in the TENO weighting strategy is adapted based on the corresponding measurement. With one set of optimal parameters, the newly proposed TENO schemes are designed to deliver excellent performance for predicting highly compressible flows with a wide range of Mach numbers. While the new very-high-order TENO schemes feature good robustness for conventional gas dynamics, the ENO-property is well preserved with the assistant of a positivity-preserving flux limiter for extreme simulations. Without loss of generality, the typical eight- and ten-point TENO-AA schemes are constructed. A set of benchmark simulations are computed to demonstrate the performance of the proposed TENO schemes.

[1]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[2]  Fan Zhang,et al.  Detonation Simulations with a Fifth-Order TENO Scheme , 2019, Communications in Computational Physics.

[3]  Omer San,et al.  Evaluation of Riemann flux solvers for WENO reconstruction schemes: Kelvin–Helmholtz instability , 2015 .

[4]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[5]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[6]  E. Hewitt,et al.  The Gibbs-Wilbraham phenomenon: An episode in fourier analysis , 1979 .

[7]  Eleuterio F. Toro,et al.  Finite-volume WENO schemes for three-dimensional conservation laws , 2004 .

[8]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[9]  Nikolaus A. Adams,et al.  An adaptive central-upwind weighted essentially non-oscillatory scheme , 2010, J. Comput. Phys..

[10]  P. Moin,et al.  Velocity transformation for compressible wall-bounded turbulent flows with and without heat transfer , 2021, Proceedings of the National Academy of Sciences.

[11]  V. Gregory Weirs,et al.  Optimization of weighted ENO schemes for DNS of compressible turbulence , 1997 .

[12]  Shock-Wave/Boundary-Layer Interactions in Transitional Rectangular Duct Flows , 2020 .

[13]  Sergio Pirozzoli,et al.  Conservative Hybrid Compact-WENO Schemes for Shock-Turbulence Interaction , 2002 .

[14]  Ivan Fedioun,et al.  Comparison of improved finite-difference WENO schemes for the implicit large eddy simulation of turbulent non-reacting and reacting high-speed shear flows , 2014 .

[15]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[16]  G. Russo,et al.  Central WENO schemes for hyperbolic systems of conservation laws , 1999 .

[17]  G. Moretti,et al.  THREE-DIMENSIONAL SUPERSONIC FLOW COMPUTATIONS , 1963 .

[18]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[19]  Chi-Wang Shu,et al.  High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments , 2016, J. Comput. Phys..

[20]  Javier Urzay,et al.  HTR solver: An open-source exascale-oriented task-based multi-GPU high-order code for hypersonic aerothermodynamics , 2020, Comput. Phys. Commun..

[21]  Yue Li,et al.  A low-dissipation shock-capturing framework with flexible nonlinear dissipation control , 2020, J. Comput. Phys..

[22]  Lin Fu,et al.  A very-high-order TENO scheme for all-speed gas dynamics and turbulence , 2019, Comput. Phys. Commun..

[23]  N. Sandham,et al.  Assessment of Low-Dissipative Shock-Capturing Schemes for the Compressible Taylor–Green Vortex , 2021 .

[24]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .

[25]  Xiangxiong Zhang,et al.  Positivity-preserving high order finite difference WENO schemes for compressible Euler equations , 2012, J. Comput. Phys..

[26]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[27]  Nikolaus A. Adams,et al.  Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws , 2017, J. Comput. Phys..

[28]  Wai-Sun Don,et al.  An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws , 2008, J. Comput. Phys..

[29]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[30]  Gabriella Puppo,et al.  Compact Central WENO Schemes for Multidimensional Conservation Laws , 1999, SIAM J. Sci. Comput..

[31]  Nikolaus A. Adams,et al.  A family of high-order targeted ENO schemes for compressible-fluid simulations , 2016, J. Comput. Phys..

[32]  Nikolaus A. Adams,et al.  Numerical symmetry-preserving techniques for low-dissipation shock-capturing schemes , 2019, Computers & Fluids.

[33]  Lin Fu,et al.  High-Order Low-Dissipation Targeted ENO Schemes for Ideal Magnetohydrodynamics , 2019, J. Sci. Comput..

[34]  G. Moretti,et al.  A time-dependent computational method for blunt body flows. , 1966 .

[35]  Steven H. Frankel,et al.  Numerical simulations of compressible multicomponent and multiphase flow using a high-order targeted ENO (TENO) finite-volume method , 2017 .

[36]  Nikolaus A. Adams,et al.  A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws , 2018, J. Comput. Phys..

[37]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[38]  D. Pullin,et al.  Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks , 2004 .

[39]  Ke Zhang,et al.  An efficient targeted ENO scheme with local adaptive dissipation for compressible flow simulation , 2021, J. Comput. Phys..

[40]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[41]  W. F. Noh.  Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux , 1985 .

[42]  Lei Luo,et al.  A sixth order hybrid finite difference scheme based on the minimized dispersion and controllable dissipation technique , 2014, J. Comput. Phys..

[43]  Yuxin Ren,et al.  A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws , 2003 .

[44]  Bruno Costa,et al.  An improved WENO-Z scheme , 2016, J. Comput. Phys..

[45]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[46]  S. Laizet,et al.  On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high‐speed compressible flows , 2020, International Journal for Numerical Methods in Fluids.

[47]  V. Gregory Weirs,et al.  A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence , 2006, J. Comput. Phys..

[48]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[49]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[50]  Guy Capdeville,et al.  A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes , 2008, J. Comput. Phys..

[51]  Sergio Pirozzoli,et al.  On the spectral properties of shock-capturing schemes , 2006, J. Comput. Phys..

[52]  Gino Moretti,et al.  Three-Dimensional Flow around Blunt Bodies , 1967 .

[53]  Jun Zhu,et al.  A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws , 2016, J. Comput. Phys..

[54]  J Korea,et al.  The Magnetohydrodynamic Kelvin-Helmholtz Instability: A Three-dimensional Study of Nonlinear Evolution , 2000, astro-ph/0008084.

[55]  Chunlei Liang,et al.  Computation of flows with shocks using the Spectral Difference method with artificial viscosity, I: Basic formulation and application , 2014 .

[56]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[57]  Chi-Wang Shu,et al.  Anti-diffusive flux corrections for high order finite difference WENO schemes , 2005 .

[58]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes , 2020, Acta Numerica.

[59]  Raphaël Loubère,et al.  A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods , 2005 .

[60]  Sergio Pirozzoli,et al.  Numerical Methods for High-Speed Flows , 2011 .

[61]  Yong-Tao Zhang,et al.  Resolution of high order WENO schemes for complicated flow structures , 2003 .

[62]  Parviz Moin,et al.  Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves , 2010, J. Comput. Phys..

[63]  Soshi Kawai,et al.  Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows , 2010, J. Comput. Phys..

[64]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[65]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection , 1977 .

[66]  Klaus A. Hoffmann,et al.  Minimizing errors from linear and nonlinear weights of WENO scheme for broadband applications with shock waves , 2013, J. Comput. Phys..

[67]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[68]  Dinshaw S. Balsara,et al.  An efficient class of WENO schemes with adaptive order , 2016, J. Comput. Phys..

[69]  N. A. Adams,et al.  Positivity-preserving flux limiters for high-order conservative schemes , 2012 .

[70]  Nikolaus A. Adams,et al.  A High-Resolution Hybrid Compact-ENO Scheme for Shock-Turbulence Interaction Problems , 1996 .

[71]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[72]  Nail K. Yamaleev,et al.  A systematic methodology for constructing high-order energy stable WENO schemes , 2009, J. Comput. Phys..

[73]  Lin Fu,et al.  A Targeted ENO Scheme as Implicit Model for Turbulent and Genuine Subgrid Scales , 2019, Communications in Computational Physics.