An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary

Abstract In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f ( u ) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f ′ ( u ) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32] . The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.

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

[2]  Ken Mattsson,et al.  A high-order accurate embedded boundary method for first order hyperbolic equations , 2017, J. Comput. Phys..

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

[4]  Semyon Tsynkov,et al.  The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes , 2012, Journal of Scientific Computing.

[5]  Chi-Wang Shu Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[6]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

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

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

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

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

[11]  Chi-Wang Shu,et al.  Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations , 2016, J. Comput. Phys..

[12]  Semyon Tsynkov,et al.  A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media , 2009, J. Comput. Phys..

[13]  ShuChi-Wang,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes, II , 1989 .

[14]  C. Peskin The immersed boundary method , 2002, Acta Numerica.

[15]  Jun Zhu,et al.  A new type of multi-resolution WENO schemes with increasingly higher order of accuracy , 2018, J. Comput. Phys..

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

[17]  Semyon Tsynkov,et al.  Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number , 2013, J. Comput. Phys..

[18]  I. Singer,et al.  High-order finite difference methods for the Helmholtz equation , 1998 .

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

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

[21]  N. K. Kulman,et al.  Method of difference potentials and its applications , 2001 .

[22]  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..

[23]  H. Kreiss,et al.  Stability Theory of Difference Approximations for Mixed Initial Boundary Value Problems. II , 1972 .

[24]  Semyon Tsynkov,et al.  Numerical Simulation of Time-Harmonic Waves in Inhomogeneous Media using Compact High Order Schemes , 2011 .

[25]  Wang Chi-Shu,et al.  Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws , 1997 .

[26]  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..

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

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

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

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

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

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

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

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