Inverse Lax–Wendroff Procedure for Numerical Boundary Treatment of Hyperbolic Equations

Abstract We discuss a high-order accurate numerical boundary condition for solving hyperbolic conservation laws on fixed Cartesian grids, while the physical domain can be arbitrarily shaped and moving. Compared with body-fitted meshes, the biggest advantage of Cartesian grids is that the grid generation is trivial. The challenge is, however, that the physical boundary does not usually coincide with grid lines. The wide stencil of the high-order interior scheme makes a stable boundary treatment even harder to realize. There are two main ingredients of this method. The first one is an inverse Lax–Wendroff procedure for inflow boundary conditions and the other one is a robust and high-order accurate extrapolation for outflow boundary conditions. This method is high-order accurate, stable, and easy to implement. It has been successfully applied to simulate interactions between compressible inviscid flows and rigid (static or moving) bodies with complex geometry.

[1]  Michele Napolitano,et al.  AN IMMERSED-BOUNDARY METHOD FOR COMPRESSIBLE VISCOUS FLOWS , 2006 .

[2]  Randall J. LeVeque,et al.  Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension , 1997, SIAM J. Sci. Comput..

[3]  Ling,et al.  NUMERICAL BOUNDARY CONDITIONS FOR THE FAST SWEEPING HIGH ORDER WENO METHODS FOR SOLVING THE EIKONAL EQUATION , 2008 .

[4]  Pietro De Palma,et al.  An immersed boundary method for compressible flows using local grid refinement , 2007, J. Comput. Phys..

[5]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

[6]  Xiangxiong Zhang,et al.  A curved boundary treatment for discontinuous Galerkin schemes solving time dependent problems , 2016, J. Comput. Phys..

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

[8]  N. Anders Petersson,et al.  A Fourth-Order Accurate Embedded Boundary Method for the Wave Equation , 2012, SIAM J. Sci. Comput..

[9]  Chi-Wang Shu,et al.  Finite Difference WENO Schemes with Lax-Wendroff-Type Time Discretizations , 2002, SIAM J. Sci. Comput..

[10]  Antonio Baeza,et al.  High Order Boundary Extrapolation Technique for Finite Difference Methods on Complex Domains with Cartesian Meshes , 2016, J. Sci. Comput..

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

[12]  Rajat Mittal,et al.  A sharp interface immersed boundary method for compressible viscous flows , 2007, J. Comput. Phys..

[13]  Lilia Krivodonova,et al.  High-order accurate implementation of solid wall boundary conditions in curved geometries , 2006 .

[14]  HEINZ-OTTO KREISS,et al.  A Second Order Accurate Embedded Boundary Method for the Wave Equation with Dirichlet Data , 2005, SIAM J. Sci. Comput..

[15]  Randall J. LeVeque,et al.  Cartesian Grid Methods for Fluid Flow in Complex Geometries , 2001 .

[16]  Nikolaus A. Adams,et al.  A conservative interface method for compressible flows , 2006, J. Comput. Phys..

[17]  Heinz-Otto Kreiss,et al.  Difference Approximations of the Neumann Problem for the Second Order Wave Equation , 2004, SIAM J. Numer. Anal..

[18]  Nail K. Yamaleev,et al.  Boundary closures for fourth-order energy stable weighted essentially non-oscillatory finite-difference schemes , 2011, J. Comput. Phys..

[19]  Cheng Wang,et al.  Efficient implementation of high order inverse Lax-Wendroff boundary treatment for conservation laws , 2012, J. Comput. Phys..

[20]  Chi-Wang Shu,et al.  Uniformly Accurate Discontinuous Galerkin Fast Sweeping Methods for Eikonal Equations , 2011, SIAM J. Sci. Comput..

[21]  S. Osher,et al.  Regular ArticleUniformly High Order Accurate Essentially Non-oscillatory Schemes, III , 1997 .

[22]  Chang Yang,et al.  An inverse Lax-Wendroff method for boundary conditions applied to Boltzmann type models , 2012, J. Comput. Phys..

[23]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[24]  M. Arienti,et al.  A level set approach to Eulerian-Lagrangian coupling , 2003 .

[25]  With Invariant Submanifolds,et al.  Systems of Conservation Laws , 2009 .

[26]  B. Sjögreen,et al.  A Cartesian embedded boundary method for hyperbolic conservation laws , 2006 .

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

[28]  Randall J. LeVeque,et al.  H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids , 2003, SIAM J. Numer. Anal..

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

[30]  J. Falcovitz,et al.  A Two-Dimensional Conservation Laws Scheme for Compressible Flows with Moving Boundaries , 1997 .

[31]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[32]  Rolf Jeltsch,et al.  A Higher-Order Boundary Treatment for Cartesian-Grid Methods , 1998 .

[33]  Jean Virieux,et al.  Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves , 2007, 0706.3825.

[34]  Eitan Tadmor,et al.  Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II , 1978 .

[35]  Keh-Ming Shyue.  A Moving-Boundary Tracking Algorithm for Inviscid Compressible Flow , 2008 .

[36]  Chi-Wang Shu,et al.  Fast Sweeping Fifth Order WENO Scheme for Static Hamilton-Jacobi Equations with Accurate Boundary Treatment , 2010, J. Sci. Comput..

[37]  Chi-Wang Shu,et al.  Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces , 2003, J. Sci. Comput..

[38]  Randall J. LeVeque,et al.  A High-Resolution Rotated Grid Method for Conservation Laws with Embedded Geometries , 2005, SIAM J. Sci. Comput..

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

[40]  E. Tadmor,et al.  Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II , 1978 .

[41]  Gino Moretti,et al.  Importance of Boundary Conditions in the Numerical Treatment of Hyperbolic Equations , 1969 .

[42]  Antonio Baeza,et al.  High Order Weighted Extrapolation for Boundary Conditions for Finite Difference Methods on Complex Domains with Cartesian Meshes , 2016, J. Sci. Comput..

[43]  Chi-Wang Shu,et al.  Development and stability analysis of the inverse Lax−Wendroff boundary treatment for central compact schemes , 2015 .

[44]  Chi-Wang Shu,et al.  A high order moving boundary treatment for compressible inviscid flows , 2011, J. Comput. Phys..

[45]  C. Peskin.  Flow patterns around heart valves: A numerical method , 1972 .

[46]  Marsha Berger,et al.  Flow Simulations on Cartesian Grids Involving Complex Moving Geometries , 1999 .

[47]  Heinz-Otto Kreiss,et al.  Difference Approximations for the Second Order Wave Equation , 2002, SIAM J. Numer. Anal..

[48]  Chi-Wang Shu,et al.  Inverse Lax–Wendroff Procedure for Numerical Boundary Conditions of Hyperbolic Equations: Survey and New Developments , 2013 .

[49]  Chi-Wang Shu,et al.  Stability analysis of the inverse Lax-Wendroff boundary treatment for high order upwind-biased finite difference schemes , 2016, J. Comput. Appl. Math..

[50]  Chi-Wang Shu,et al.  Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws , 2010, J. Comput. Phys..

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