Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems

Recent progress in the development of a class of low dissipative high order (fourth-order or higher) filter schemes for multiscale Navier-Stokes, and ideal and non-ideal magnetohydrodynamics (MHD) systems is described. The four main features of this class of schemes are: (a) multiresolution wavelet decomposition of the computed flow data as sensors for adaptive numerical dissipative control, (b) multistep filter to accommodate efficient application of different numerical dissipation models and different spatial high order base schemes, (c) a unique idea in solving the ideal conservative MHD system (a non-strictly hyperbolic conservation law) without having to deal with an incomplete eigensystem set while at the same time ensuring that correct shock speeds and locations are computed, and (d) minimization of the divergence of the magnetic field (@?.B) numerical error. By design, the flow sensors, different choice of high order base schemes and numerical dissipation models are stand-alone modules. A whole class of low dissipative high order schemes can be derived at ease, making the resulting computer software very flexible with widely applicable. Performance of multiscale and multiphysics test cases are illustrated with many levels of grid refinement and comparison with commonly used schemes in the literature.

[1]  C. Wilke.  A Viscosity Equation for Gas Mixtures , 1950 .

[2]  H. C. Yee,et al.  Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computations , 2001 .

[3]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[4]  Björn Sjögreen,et al.  Numerical experiments with the multiresolution scheme for the compressible Euler equations , 1995 .

[5]  H. C. Yee,et al.  Extension of Efficient Low Dissipative High Order Schemes for 3-D Curvilinear Moving Grids , 2000 .

[6]  H. C. Yee,et al.  Designing Adaptive Low-Dissipative High Order Schemes for Long-Time Integrations. Chapter 1 , 2001 .

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

[8]  Ravi Samtaney,et al.  Suppression of the Richtmyer–Meshkov instability in the presence of a magnetic field , 2003 .

[9]  Margot Gerritsen,et al.  Designing an efficient solution strategy for fluid flows. 1. A stable high order finite difference scheme and sharp shock resolution for the Euler equations , 1996 .

[10]  Bjorn Sjogreen,et al.  Divergence Free High Order Filter Methods for the Compressible MHD Equations * , 2003 .

[11]  Stéphane Mallat,et al.  Characterization of Signals from Multiscale Edges , 2011, IEEE Trans. Pattern Anal. Mach. Intell..

[12]  Chi-Wang Shu,et al.  A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes , 2006, J. Comput. Phys..

[13]  Björn Sjögreen,et al.  Grid convergence of high order methods for multiscale complex unsteady viscous compressible flows , 2001 .

[14]  Björn Sjögreen,et al.  Low dissipative high‐order numerical simulations of supersonic reactive flows , 2003 .

[15]  J. Hawley,et al.  Simulation of magnetohydrodynamic flows: A Constrained transport method , 1988 .

[16]  Stéphane Mallat,et al.  Multifrequency channel decompositions of images and wavelet models , 1989, IEEE Trans. Acoust. Speech Signal Process..

[17]  Christopher J. Roy,et al.  Evaluation of Detached Eddy Simulation for Turbulent Wake Applications , 2005 .

[18]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[19]  Björn Sjögreen,et al.  Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computations , 2001 .

[20]  David Gottlieb,et al.  Spectral Simulation of Supersonic Reactive Flows , 1998 .

[21]  B. Mcbride,et al.  Thermodynamic Properties to 6000K for 210 Substances Involving the First 18 Elements. NASA SP-3001 , 1963 .

[22]  P. Olsson.  Summation by parts, projections, and stability. II , 1995 .

[23]  Björn Sjögreen,et al.  Performance of High Order Filter Methods for a Richtmyer-Meshkov Instability , 2009 .

[24]  Gilbert Strang,et al.  Wavelets and Dilation Equations: A Brief Introduction , 1989, SIAM Rev..

[25]  Paul R. Woodward,et al.  A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations , 1998 .

[26]  B. Mcbride,et al.  THERMODYNAMIC PROPERTIES TO 6000 K FOR 210 SUBSTANCES INVOLVING THE FIRST 18 ELEMENTS , 1963 .

[27]  Jan,et al.  Boundary and Interface Conditions for High Order Finite Difference Methods Applied to the Euler and Navier-Stokes Equations , 2022 .

[28]  Björn Sjögreen,et al.  Simulation of Richtmyer–Meshkov instability by sixth-order filter methods , 2007 .

[29]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[30]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[31]  H. C. Yee,et al.  Entropy Splitting and Numerical Dissipation , 2000 .

[32]  Helen C. Yee,et al.  Efficient Low Dissipative High Order Schemes for Multiscale MHD Flows , 2002 .

[33]  Stéphane Mallat,et al.  Singularity detection and processing with wavelets , 1992, IEEE Trans. Inf. Theory.

[34]  G. Strang.  Wavelet transforms versus Fourier transforms , 1993, math/9304214.

[35]  H. C. Yee,et al.  Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence , 2002 .

[36]  Björn Sjögreen,et al.  Divergence Free High Order Filter Methods for the Compressible MHD Equations , 2003, HPSC.

[37]  R. Svehla,et al.  Estimated Viscosities and Thermal Conductivities of Gases at High Temperatures , 1962 .

[38]  Björn Sjögreen,et al.  Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for High Order Methods , 2004, J. Sci. Comput..

[39]  H. C. Yee,et al.  A Numerical Study of Resistivity and Hall Effects for a Compressible MHD Model , 2005 .

[40]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[41]  Björn Sjögreen,et al.  Non-Linear Filtering and Limiting in High Order Methods for Ideal and Non-Ideal MHD , 2006, J. Sci. Comput..

[42]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[43]  Neil D. Sandham,et al.  Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters , 1999 .

[44]  Björn Sjögreen,et al.  Nonlinear filtering in compact high-order schemes , 2006, Journal of Plasma Physics.

[45]  Ravi Samtaney,et al.  Regular shock refraction at an oblique planar density interface in magnetohydrodynamics , 2005, Journal of Fluid Mechanics.

[46]  H. C. Yee,et al.  A class of high resolution explicit and implicit shock-capturing methods , 1989 .

[47]  Miguel R. Visbal,et al.  Further development of a Navier-Stokes solution procedure based on higher-order formulas , 1999 .

[48]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[49]  R. D. Richtmyer.  Taylor instability in shock acceleration of compressible fluids , 1960 .

[50]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[51]  Kenneth G. Powell,et al.  AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension) , 1994 .

[52]  C. Basdevant,et al.  Wavelet spectra compared to Fourier spectra , 1995 .

[53]  Björn Sjögreen,et al.  High order centered difference methods for the compressible Navier-Stokes equations , 1995 .

[54]  Wai Sun Don,et al.  Numerical simulation of shock-cylinder interactions I.: resolution , 1995 .