Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of. Conservation Laws 1. Abstract

Transcription

1 Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of Conservation Laws Sirui Tan and Chi-Wang Shu 3 Abstract We develop a high order finite difference numerical boundary condition for solving hyperbolic conservation laws on a Cartesian mesh. The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary intersects the grids in an arbitrary fashion. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly use the partial differential equation to write the normal derivatives to the inflow boundary in terms of the time derivatives and the tangential derivatives. With these normal derivatives, we can then impose accurate values of ghost points near the boundary by a Taylor expansion. At outflow boundaries, we use Lagrange extrapolation or least squares extrapolation if the solution is smooth, or a weighted essentially non-oscillatory (WENO) type extrapolation if a shock is close to the boundary. Extensive numerical examples are provided to illustrate that our method is high order accurate and has good performance when applied to one and two dimensional scalar or system cases with the physical boundary not aligned with the grids and with various boundary conditions including the solid wall boundary condition. AMS subject classification: 65M06 Keywords: numerical boundary conditions; hyperbolic conservation laws; finite difference method; Cartesian mesh; Lax-Wendroff procedure; extrapolation; solid wall Research supported by AFOSR grant FA and NSF grant DMS Division of Applied Mathematics, Brown University, Providence, RI sirui@dam.brown.edu 3 Division of Applied Mathematics, Brown University, Providence, RI shu@dam.brown.edu

2 Introduction In this paper we develop a high order boundary treatment for solving strongly hyperbolic conservation laws with high order finite difference methods on a Cartesian mesh. In computational domain with complex geometries, a Cartesian mesh makes the numerical method efficient and easy to implement compared with boundary fitted structured or unstructured meshes. However, there are two kinds of difficulties imposing inflow boundary conditions. First, a high order interior scheme needs a suitable treatment for several ghost points near the boundary because of the wide numerical stencil. Secondly, the grid points are usually not located on the physical boundary when using a Cartesian mesh. In other words, the boundary intersects the grids in an arbitrary fashion. In finite volume methods, this leads to a severe time step restriction. The so-called h-box method is developed to overcome this problem, see [] and the references therein. In solving compressible inviscid Euler equations, the most popular way to impose the no-penetration boundary condition at solid walls is the reflection technique, where ghost points are added behind the wall. All interior solution components are reflected symmetrically to values of ghost points except for the normal velocity whose sign is reversed. This method works well if the grid points are symmetrically located with respect to the wall and leads to large errors otherwise. An accurate implementation of solid wall boundary conditions in curved geometries is developed for discontinuous Galerkin methods on unstructured, straight-sided element meshes in []. A Cartesian embedded boundary method based on the finite difference formulation is developed to solve the wave equation with Dirichlet or Neumann boundary conditions in [9, 0, ] and hyperbolic conservation laws in [7]. The technique used in [7] is formally second order accurate with an explicit time step determined by the grid size away from the boundary. It is essentially based on a three point interior scheme so that only points just outside the boundary are ghost points, which are imposed by extrapolation. The feasibility and effectiveness to generalize this approach to higher order remain to be demonstrated. The idea of the inverse Lax-Wendroff procedure is first introduced in [7] for solving static

3 Hamilton-Jacobi equations with a third order method. Later, Xiong et al. extend it to fifth order for the same type of problems in [0]. This approach is based on repeatedly using the partial differential equation (PDE) to write the normal derivatives to the inflow boundary in terms of the tangential derivatives of the given boundary condition. With these normal derivatives, we can obtain accurate values of ghost points by a Taylor expansion from a point located on the boundary. In this paper, we systematically extend the inverse Lax-Wendroff type procedure for solving hyperbolic conservation laws. The difference with [7, 0] mainly relies on three points. First, for time dependent problems, our boundary treatment procedure is in essence repeatedly using the PDE to convert normal spatial derivatives to tangential and time derivatives of the given boundary condition. Secondly, in systems of conservation laws, the inflow and outflow boundary conditions are coupled. As a result, special care must be taken for imposing outflow boundary conditions. We use high order Lagrange extrapolation or least squares extrapolation if the solution is smooth. In the presence of shocks near the outflow boundary, we develop a weighted essentially non-oscillatory (WENO) type extrapolation to prevent oscillations and maintain accuracy. Finally, this method should work well for solid wall boundary conditions. Our boundary treatment procedure, which uses repeatedly the PDE to convert normal spatial derivatives to tangential and time derivatives of the given boundary condition, is in some sense an inverse to the usual Lax-Wendroff procedure [3], in which the PDE is repeated used to convert time derivatives to spatial derivatives when discretizing the PDE in time with high order accuracy. We therefore refer to our method as the inverse Lax-Wendroff procedure. This paper is organized as follows. In Section, we first give an overview of the discretization of the problem. Then we illustrate the idea of the inverse Lax-Wendroff procedure by one-dimensional scalar conservation laws. We will see a need for robust extrapolation if there is a shock near the outflow boundary. We develop a WENO type extrapolation for this purpose. The linear stability of our numerical boundary condition is shown afterwards. The 3

4 method for one-dimensional scalar equation can be generalized to one-dimensional systems and further to two-dimensional problems. In Section 3, a variety of numerical examples are provided to demonstrate the effectiveness and generality of our approach. Concluding remarks are given in Section 4. Scheme formulation We consider strongly hyperbolic conservation laws for U = U(x, y, t) { U t + F (U) x + G(U) y = 0 (x, y) Ω, t > 0, U(x, y, 0) = U 0 (x, y) (x, y) Ω, (.) on a bounded domain Ω with appropriate boundary conditions prescribed on Ω at time t. We assume Ω is covered by a uniform Cartesian mesh Ω h = {(x i, y j ) : 0 i N x, 0 j N y } with mesh size x = y. The semi-discrete approximation of (.) is given by d dt U i,j(t) = ( ˆF i+/,j x ˆF ) i /,j (Ĝi,j+/ Ĝi,j /), (.) y where ˆF i+/,j and Ĝi,j+/ are numerical fluxes. We use a third order total variation diminishing (TVD) Runge-Kutta method [6] to integrate the system of ordinary differential equations (ODEs) (.) in time. For ease of notations, we suppose a system of initial value problems of ODEs is written as u t = L(u). The third order TVD Runge-Kutta method is given by u () = u n + tl(u n ), u () = 3 4 un + 4 u() + 4 tl(u() ), (.3) u n+ = 3 un + 3 u() + 3 tl(u() ). To make the accuracy in time compatible with the accuracy in space, we take t = O ( ) x s 3 if the space discretization is of order s, when performing the accuracy tests. Other time 4

5 discretization techniques, such as the various strong stability preserving Runge-Kutta or multistep methods [4], could of course be used as well. Special care must be taken when we impose time dependent boundary conditions in the two interior stages of the Runge-Kutta method (.3). The traditional match of time u n g(t n ), u () g(t n + t), u () g(t n + t/), decreases the accuracy to second order as pointed out in []. It is shown in [] that for hyperbolic systems (.), the following match of time maintains the third order accuracy of (.3) u n g(t n ), u () g(t n ) + tg (t n ), (.4) u () g(t n ) + tg (t n ) + 4 t g (t n ). For simplicity, we denote the boundary conditions for all stages at t = t n by g(t n ), although g(t n ) is actually different for each stage according to (.4). For definiteness, we use the fifth order finite difference WENO scheme [8] to form the numerical fluxes ˆF i+/,j and Ĝi,j+/ in (.), although our method is independent of the interior scheme. The fifth order WENO scheme requires a seven point stencil in both x and y directions, which is much wider compared with low order schemes. Near Ω where the numerical stencil is partially outside of Ω, up to three ghost points are needed in each direction. We concentrate on how to define the values of U i,j at ghost points in the rest of the paper. 5

6 . One-dimensional scalar conservation laws To best illustrate the idea of the inverse Lax-Wendroff type procedure, we use D scalar conservation laws as an example u t + f(u) x = 0 x (, ), t > 0, u(, t) = g(t) t > 0, u(x, 0) = u 0 (x) x [, ]. We assume f (u(, t)) α > 0 and f (u(, t)) α > 0 for t > 0. (.5) This assumption guarantees the left boundary x = is an inflow boundary where a boundary condition is needed and the right boundary x = is an outflow boundary where no boundary condition is needed. Let us discretize the interval (, ) by a uniform mesh + x/ = x 0 < x < < x N = x/. (.6) Notice that both x 0 and x N are not located on the boundary, which is chosen this way on purpose since it is usually not possible to align boundary with grid points in a twodimensional domain with complex geometries. At the inflow boundary x =, a Taylor expansion of order s gives u(x j, t n ) = s k=0 (x j + ) k k u + O( x s ), k! x k x=,t=t n for j =,, 3. Hence a s-th order approximation of the values u j at the ghost points is s (x j + ) k k u u j =, j =,, 3. (.7) k! x k k=0 x=,t=t n Here we suppress the t n dependence on the left hand side. We already have u(, t n ) = g(t n ). To obtain the spatial derivatives, we utilize the PDE u t + f (u)u x = 0 and evaluate it at x =, t = t n. We have u x (, t n ) = u t(, t n ) f (u(, t n )) 6

7 = g (t n ) f (g(t n )), where f (g(t n )) is bounded away from zero by the assumption that x = is an inflow boundary. Differentiating the PDE with respect to time yields u tt + f (u)u t u x + f (u)u xt = 0. (.8) The term u xt can be written as u xt = (u t ) x = (f (u)u x ) x = f (u)u x f (u)u xx. Substituting it into (.8), we obtain an equation for u xx u tt + f (u)u t u x f (u)f (u)u x f (u) u xx = 0. (.9) Solving (.9) for u xx and evaluating it at x =, t = t n, we have u xx (, t n ) = g (t n ) + f (g(t n ))g (t n )u x (, t n ) f (g(t n ))f (g(t n ))u x (, t n) f (g(t n )) = f (g(t n ))g (t n ) f (g(t n ))g (t n ) f (g(t n )) 3. Following the same procedure, we can obtain values of k u x k x=,t=tn, k =,, s. The idea of converting time derivatives to spatial derivatives by repeatedly using the PDE comes from the original Lax-Wendroff scheme [3]. Since we convert spatial derivatives to time derivatives instead, our method is called the inverse Lax-Wendroff procedure. We remark that this procedure is independent of the interior scheme and the location of the boundary. The time derivatives can be obtained by either using the analytical derivatives of g(t) if available or numerical differentiation. In the case of shocks going through the boundary, g(t) is discontinuous. The stencil used for numerical differentiation should not contain any discontinuity, e.g. an essentially non-oscillatory (ENO) procedure [6] can be used for this numerical differentiation. 7

8 At the outflow boundary x =, extrapolation of appropriate order is used. The s-th order extrapolation is given by s ( ) s ( ) k u k j k = 0, j = N +, N +, N + 3. (.0) k=0 For example, the fifth order extrapolation is u j = u j 5 5u j 4 + 0u j 3 0u j + 5u j, j = N +, N +, N + 3. An equivalent way to do the extrapolation is to use a Taylor expansion u j = s k=0 (x j ) k u (k), (.) k! where u (k) is a (s k)-th order approximation of k u x k x=,t=tn. Notice that (.) is consistent with (.7), which is useful in the system case discussed later. boundary, u (k) can be easily obtained by u (k) = dk p s (x) dx k If u is smooth near the, (.) x= where p s (x) is a Lagrange polynomial of degree s satisfying p s (x N i ) = u N i, i = 0,, s. When a shock goes out of the boundary, there are not enough points between the shock and the boundary for high order extrapolation at a particular time. In this situation, high order extrapolation may lead to a severe oscillation near the shock. To prevent this from happening, we would like to have a lower order accurate but more robust extrapolation. The WENO type extrapolation is developed for this purpose.. One-dimensional WENO type extrapolation Assume that we have a stencil of three points x 0 = 0, x = x, x = x with point values u j at x j, j = 0,,. We aim to obtain a (3 k)-th order approximation of d k u dx k x= x/, which is denoted by u (k), k = 0,,. We have three candidate substencils given by S r = {x 0,, x r }, r = 0,,. 8

9 On each substencil S r, we can easily construct a Lagrange polynomial p r (x) of degree r p 0 (x) = u 0, p (x) = u u 0 x x + u 0, p (x) = u 0 u + u x + 3u 0 + 4u u x + u x 0. x Suppose u(x) is smooth on S, u (k) can be extrapolated by u (k) = r=0 where d 0 = x, d = x, d = x x. d r d k p r (x) dx k, x= x/ We now look for WENO type extrapolation in the form u (k) = r=0 ω r d k p r (x) dx k, (.3) x= x/ where ω r are the nonlinear weights depending on the values of u j. In the case that u(x) is smooth in S, we would like to have ω 0 = O( x ), ω = O( x), (.4) ω = ω 0 ω. This implies (.3) is (3 k)-th order accurate. Following the idea of the usual WENO procedure in [8], the nonlinear weights ω r are chosen to be ω r = α r s=0 α s with α r = d r (ɛ + β r ), where ɛ = 0 6 and β r are the smoothness indicators, which are determined by β 0 = x, 9

10 β = l= 0 x ( ) d x l l dx p (x) dx l = (u u 0 ), (.5) 0 ( ) d β = x l l dx p (x) dx l l= x = (6u u + 74u 0u + 5u 96u u 0 + 4u u )/. (.6) We next show that (.4) is achieved by this choice of β r. Taylor expansion of (.5) and (.6) at x = 0 gives ( β = u x + u u x u + ) 3 u u x 4 + O( x 5 ), (.7) ( 4 β = u x u u x u 5 ) 3 u u x 4 + O( x 5 ). (.8) We omit ɛ in the expression of α r and do the following estimates with the assumption that C = (u ) 4 0. α 0 = x x, 4 x α = x 4 (C + O( x)) = x ( ) + O( x), x 4 C α = + O( x) x 4 ω 0 = ( C + O( x) ) ; x ( ) ( ) x + x C + O( x) + ( + O( x)) C + O( x) = x C + O( x) = C x + O( x 3 ), ( ) ω = x C + O( x) C + O( x) = x + O( x ). If S contains a discontinuity, we have β = O() and β = O(). u (0) reduces to a first order approximation since α 0 = x, 0

11 α = O( x), α = O(); ω = O( x) + O( x) + O() x = O( x 3 ), ω = O() + O( x) + O() x = O( x ). Notice that as x 0 the weights assigned to the non-smooth stencils S and S vanish in a rate of O( x 3 ) and O( x ), respectively. If only S contains a discontinuity, we have β = O(). u (0) is a second order approximation since α 0 = α = α = O(); ω 0 = ω = x, ( ) + O( x) x 3 C ( + x x 3 x = C x + O( x ),, C + O( x) O() ( + x x 3 C + O( x) = C x 3 + O( x 4 ). ) + O() ) + O() If u = 0 but u 0, then (.7) and (.8) reduce to β = 4 u x 4 + O( x 5 ), β = 4 3 u x 4 + O( x 5 ). The above estimates can be done in a similar fashion. We only show (.4) here. Setting C = (u ) 4, we readily check α 0 = x x 4,

12 α = x C 6 x 8 + O( x 9 ) = x x 8 α = + O( x) x 8 ω 0 = ω =.3 Linear stability ( ) 6 + O( x), C x + x x 4 ( ( 9 6C + O( x) x ) 6 C + O( x) = 6C 9 x6 + O( x 7 ), x + x x 4 ( = 56 9 x + O( x ). ( x 6 x 4 ) ; ) C + O( x) ) 6 C + O( x) + +O( x) ( + +O( x) 9 x 4 6C + O( x) x 4 ( ) ) 9 6C + O( x) In this section, we prove the stability of our numerical boundary conditions for linear wave equations according to the theory of Gustafsson, Kreiss and Sundström [5] (henceforth GKS). For semi-discrete schemes, the GKS stability is studied in Strikwerda [8]. We use Strikwerda s procedure as a validation tool here. We consider quarter plane problems u t + u x = 0 x (0, + ), t > 0, u(0, t) = g(t) = 0 t > 0, u(x, 0) = u 0 (x) x [0, + ), and { ut u x = 0 x (0, + ), t > 0, u(x, 0) = u 0 (x) x [0, + ), (.9) (.0) for inflow boundary conditions and outflow boundary conditions, respectively. The stability of each individual case insures the stability of our method on a bounded domain. We assume the dissipative semi-discrete interior scheme has the form du j = p a i u j+i, dt x j 0. (.) i= r For example, the fifth order WENO scheme with linear weights applied to (.9) reads du j = ( dt x 30 u j 3 4 u j + u j 3 u j u j+ + ) 0 u j+, j 0. (.)

13 To solve (.9), we use the inverse Lax-Wendroff type procedure to impose the inflow boundary condition by u ν = 0, ν =,, r. (.3) An eigenvalue problem is formed by assuming a solution of the form u(x j, t) = u j e φt. The resolvent equation for (.) is xφu j = p a i u j+i. i= r To solve this difference equation, we set u j = κ j u 0 and obtain the characteristic equation xφ = p a i κ i. (.4) i= r For Re(φ) > 0, we assume κ i, i =,, q, are all the distinct roots of (.4) which satisfies κ i <, each with multiplicity γ i. A general candidate eigensolution has the form u j = q γ i α i,k j k (κ i (φ)) j. i= k= The boundary condition (.3) yields a linear system of equations for α i,k γ q i ν k κ ν i α i,k = 0, ν =,, r. (.5) i= k= Notice that (.5) is a square system since q γ i = r. i= Denoting the coefficient matrix by E, it can be shown, by elementary column operations, that E reduces to a generalized Vandermonde matrix. Thus [ q ] [ ] det E = C (κ i κ j ) γ iγ j, i= κ ρ i i i<j q where C is a constant and ρ i are integers. Since κ i are distinct, det E 0. Therefore, there are no nontrivial eigensolutions and our semi-discrete scheme (.) with boundary condition (.3) is stable. 3

14 For the outflow boundary condition in (.0), we can show that the semi-discrete problem (.) with the extrapolation (.0) is stable for all s. The proof is essentially covered in [3] where a fully discrete scheme is analyzed. We finally remark that the time step restriction of solving the system of ODEs (.) with our boundary treatment is not more severe than the pure initial value problem according to our computational experience. The standard CFL conditions determined by the interior schemes are used in the numerical examples in Section 3..4 One-dimensional systems We consider D compressible Euler equations U t + F (U) x = 0, x (, ), t > 0, where the conservative variables U = U U U 3 = ρ ρu E and the flux F (U) = U U ) U U U U U (γ ( )U γ γu 3 γ = ρu ρu + p u(e + p), with appropriate boundary conditions and initial conditions. As is customary, ρ, u, p and E describe the density, velocity, pressure and total energy, respectively. The equation of state has the form E = p γ + ρu, where γ =.4 for air at ordinary temperatures. We consider the right boundary x =. The left boundary can be treated in a similar fashion. To decide the inflow and outflow boundary conditions at time t, we need a local characteristic decomposition of the PDEs. At the boundary, we denote the Jacobian matrix of the 4

15 flux by A (U b ) = F (U) U, U=U b where U b = U(, t). A (U b ) has three eigenvalues λ (U b ) = u b c b, λ (U b ) = u b, λ 3 (U b ) = u b + c b and a complete set of left eigenvectors l (U b ), l (U b ), l 3 (U b ) which forms a matrix l (U b ) l, (U b ) l, (U b ) l,3 (U b ) L(U b ) = l (U b ) = l, (U b ) l, (U b ) l,3 (U b ). l 3 (U b ) l 3, (U b ) l 3, (U b ) l 3,3 (U b ) As in the scalar case, we assume at present that all the eigenvalues are bounded away from zero. The number of boundary conditions depends on the number of nonpositive eigenvalues. For simplicity, we assume that U m (, t) = g m (t), m p is prescribed at the boundary if λ m (U b ) < 0, m p. Indeed, this is equivalent to prescribing the ingoing characteristic variable V I as a function of the outgoing characteristic variable V, where V I is the first p components of V = L(U b )U and V is the last 3 p components of V. With the local characteristic decomposition, our problem reduces to three scalar conservation laws in terms of the characteristic variable V. In our numerical scheme, we use the inverse Lax-Wendroff procedure for inflow boundary conditions of V I and extrapolation for outflow boundary conditions of V. To illustrate how this works, we assume λ (U b ) λ (U b ) < 0 and λ 3 (U b ) > 0 at the boundary. Thus an appropriate set of boundary conditions is U (, t) = g (t), U (, t) = g (t), for t > 0. We still use the uniform mesh described in (.6). At time level t n, we assume U j, j = 0,, N, have been updated by the interior scheme. The values of ghost points are approximated by a (s )-th order Taylor expansion (U m ) j = s k=0 (x j ) k k! U (k) m, m =,, 3, j = N +, N +, N + 3, (.6) 5

16 where U (k) m is a (s k)-th order approximation of the spatial derivatives k U m (, t x k n ). In the local characteristic decomposition, we replace U b by U N since U(, t n ) is not always prescribed. We define the outgoing local characteristic variable V 3 at grid points near the boundary by (V 3 ) j = l 3 (U N )U j, j = N 4,, N. (.7) We extrapolate (V 3 ) j to the boundary either with (.) if the solution is smooth near the boundary or with the WENO type extrapolation (.3) if a shock is close to the boundary. We denote the extrapolated k-th order derivatives of V 3 at the boundary by V (k) 3, k = 0,, s. Obviously, we impose U (0) = g (t n ) and U (0) = g (t n ). U (0) 3 is obtained by U (0) 3 = [ ] V (0) 3 l 3, (U N )U (0) l 3, (U N )U (0). l 3,3 (U N ) Next we try to find the spatial derivatives U () m with the inverse Lax-Wendroff procedure for U and U, together with the extrapolation of V 3. The first two equations of the Euler system and the extrapolated value V () 3 give us (γ )U () γ U () = g (t n ), ( ) (0) U U () U (0) U (0) U () U (0) = g (t n ), (.8) l 3, (U N )U () + l 3, (U N )U () + l 3,3 (U N )U () 3 = V () 3. Solving this linear system, we obtain first order derivatives U () m. We remark that the condition number of the coefficient matrix in (.8) is small with the assumption that all the eigenvalues are bounded away from zero. Repeatedly using the Euler equations, we are able to form a linear system with the spatial derivatives U (k) m as the unknowns and the right-hand side depending on derivatives of g, g, extrapolated derivatives of V 3 and lower order spatial derivatives Um (l), l = 0,, k. The coefficient matrix always depends only on Um (0). Before we summarize our algorithm, we turn to the issue of zero eigenvalues. For simplicity, we assume λ (U N ) is close to zero, or the Mach number satisfies β M N = u N /c N β, 6

17 where β is a small positive number, say β = 0.0. In this situation, the local characteristic variable V can either go into the boundary or go out of the boundary. On the one hand, if we assume an ingoing V, we still have the linear system (.8), which is however ill-conditioned. The condition number will be larger and larger as M N gets close to zero. To make the system well-conditioned, we add to (.8) the equation of extrapolation of V l, (U N )U () + l, (U N )U () + l,3 (U N )U () 3 = V (). (.9) (.8) and (.9) form a system of four equations with three unknowns. We consider it as a linear least squares problem. The condition number of the 4 3 matrix is small. On the other hand, if only one boundary condition is prescribed, such as the case of solid wall boundary condition u = 0, we have to assume an outgoing V. Now the second equation of (.8) is replaced by (.9) and the resulting system is well-conditioned as well. We summarize our algorithm for imposing the values of U j at ghost points near the right boundary x = as follows, assuming that U j, j = 0,, N, have been updated at time level t n.. Compute the eigenvalues λ m (U N ) and left eigenvectors l m (U N ) of the Jacobian matrix A (U N ) for m =,, 3. Decide the prescribed inflow boundary conditions g m (t) according to the signs of λ m (U N ).. Form the outgoing characteristic variables (V m ) j, j = N 4,, N, as in (.7). Extrapolate (V m ) j to the boundary to obtain Vm (k), k = 0,, s, with Lagrange extrapolation (.) or the WENO type extrapolation (.3). 3. Solve for Um (0), m =,, 3, with the prescribed boundary conditions and extrapolated values V (0) m. 4. For k =,, s, use the inverse Lax-Wendroff type procedure to write the k-th order derivatives of g m (t) as a linear combination of k-th order spatial derivatives and other terms. Together with the extrapolation equations, form a linear system (such as (.8)) 7

18 or a linear least squares problem (such as (.8) and (.9)) for Um (k), m =,, 3. Solve for U (k) m, m =,, Impose the values of the ghost points by the Taylor expansion (.6)..5 Two-dimensional case The approach can be easily generalized to two-dimensional problems (.). We consider D compressible Euler equations U t + F (U) x + G(U) y = 0, (x, y) Ω, t > 0, (.30) where U = U U U 3 U 4 = ρ ρu ρv E, F (U) = G(U) = ρu ρu + p ρuv u(e + p) ρv ρuv ρv + p v(e + p),, with appropriate boundary conditions and initial conditions. ρ, u, v, p and E describe the density, x-velocity, y-velocity, pressure and total energy, respectively. The equation of state has the form E = p γ + ρ(u + v ), where γ =.4 for air at ordinary temperatures. We assume the values of the grid points inside domain Ω have been updated by the interior scheme. To define the value of U i,j at a ghost point P = (x i, y j ), we find a point 8

19 P 0 = (x 0, y 0 ) on the boundary Ω so that the outward normal n = (n, n ) to Ω at P 0 goes through P. We set up a local coordinate system at P 0 by ( ˆx ŷ ) ( cos θ sin θ = sin θ cos θ ( ) x = T, y ) ( x y ) (.3) where θ is the angle between the normal n and the x-axis and T is a rotation matrix. Notice that the ˆx-axis points in the normal direction to Ω at P 0 and the ŷ-axis points in the tangential direction to Ω at P 0. In this local coordinate system, the Euler system (.30) is Û t + F (Û) x + G(Û) y = 0, (.3) where Û = Û Û Û 3 Û 4 = ρ ρû ρˆv E, ( û ˆv ) = T ( u v ). At the boundary, we have the Jacobian matrix of the normal flux F (Û) A (Û b) =, Û Û= Û b where Û b = Û(x 0, y 0, t). A (Û b) has four eigenvalues λ (Û b) = û b c b, λ (Û b) = λ 3 (Û b) = û b, λ 4 (Û b) = û b + c b and a complete set of left eigenvectors l (Û b), l (Û b), l 3 (Û b), l 4 (Û b) which forms a matrix L(Û b) = l (Û b) l (Û b) l 3 (Û b) l 4 (Û b) = l, (Û b) l, (Û b) l,3 (Û b) l,4 (Û b) l, (Û b) l, (Û b) l,3 (Û b) l,4 (Û b) l 3, (Û b) l 3, (Û b) l 3,3 (Û b) l 3,4 (Û b) l 4, (Û b) l 4, (Û b) l 4,3 (Û b) l 4,4 (Û b). Now the number of prescribed boundary conditions depends on the signs of λ m (Û b), m =,, 4. For simplicity, we assume that λ m (Û b) α < 0, m =,, 3 and λ 4 (Û b) α > 0. 9

20 The boundary conditions are Û (x 0, y 0, t) = g (t), Û (x 0, y 0, t) = g (t), Û 3 (x 0, y 0, t) = g 3 (t), for t > 0. The value of Û i,j at ghost point P is approximated by a (s )-th order Taylor expansion (Ûm) i,j = s k=0 d k k! Û m (k), m =,, 4, (.33) where d is the ˆx-coordinate of P and Û (k) m is a (s k)-th order approximation of the normal derivatives kûm ˆx k (x 0, y 0, t n ). In the local characteristic decomposition, we replace Û b by Û 0 which is the value of a grid point nearest to P 0 among all the grid points inside Ω. We define the outgoing local characteristic variable V 4 at grid points near P 0 by (V 4 ) µ,ν = l 4 (Û 0)Û µ,ν, (x µ, y ν ) E i,j, (.34) where E i,j is a set of grid points inside Ω which are used to construct an extrapolating polynomial. The construction of E i,j will be discussed in Section.6. We extrapolate (V 4 ) µ,ν to the boundary and denote the extrapolated ˆx-derivatives of V 4 at the boundary by V (k) 4, k = 0,, s. Obviously, we impose Û (0) = g (t n ), Û (0) = g (t n ) and Û (0) 3 = g 3 (t n ). Û (0) 4 is obtained by Û (0) 4 = l 4,4 (Û 0) [ ] V (0) 4 l 4, (Û 0)Û (0) l 4, (Û 0)Û (0) l 4,3 (Û 0)Û (0) 3. Next we try to find the ˆx-derivatives Û () m with the inverse Lax-Wendroff procedure for Û, Û, Û3, together with the extrapolation of V 4. The first three equations of (.3) and the extrapolated value V () 4 give us AÛ () = b, (.35) 0

21 where A = γ 3 ( U (0) U (0) ( ) U (0) 3 (3 γ) U (0) U (0) ( γ) U (0) U (0) 3 (γ ) U (0) ) + γ U (0) U (0) U (0) 3 U (0) U (0) 3 U (0) U (0) U (0) l 4, (Û 0) l 4, (Û 0) l 4,3 (Û 0) l 4,4 (Û 0) 0, Û () = Û () Û () Û () 3 Û () 4 and b = g 3 (t n ) ŷ g (t n ) ŷ ( g (t n ) ŷ [ ) (0) (Û 3 Û (0) (γ )Û (0) 4 γ V () 4 ) Û (0) 3 Û (0) (0) Û Û (0) + 3 γ (0) Û 3 Û (0) ]. Solving this 4 4 linear system, we obtain first order normal derivatives Û m (). Again, the condition number of A in (.35) is small with the assumption that all the eigenvalues are bounded away from zero. Repeatedly using the Euler equations, we are able to form a linear system with the normal derivatives Û (k) m as the unknowns and the right-hand side depending on derivatives of g, g, g 3, tangential derivatives, extrapolated normal derivatives of V 4 and lower order normal derivatives Û m (l), l = 0,, k. The coefficient matrix depends only on Û (0) m. To compute the tangential derivatives in b, we use either analytical expressions if available or numerical differentiation. The later is always possible since we have obtained the values of Û (0) m along the boundary. For robustness, a least squares polynomial of suitable degree is used if the solution is smooth near the boundary, or WENO type differentiation is used otherwise.

22 We now summarize our algorithm for imposing the values of U i,j at ghost points as follows, assuming the values of the grid points inside Ω have been updated at time level t n.. For each ghost point (x i, y j ), we do the following three steps: Decide the local coordinate system (.3). Compute the eigenvalues λ m (Û 0) and left eigenvectors l m (Û 0) of the Jacobian matrix A (Û 0) for m =,, 4. Decide the prescribed inflow boundary conditions g m (t) according to the signs of λ m (Û 0). Form the outgoing characteristic variables (V m ) µ,ν, (x µ, y ν ) E i,j as in (.34). Extrapolate (V m ) µ,ν to the boundary to obtain Vm (k), k = 0,, s, with a Lagrange polynomial or a least squares polynomial if the solution is smooth near the boundary, or with the WENO type extrapolation otherwise. Details of the two-dimensional extrapolation will be discussed in Section.6. Solve for Û m (0), m =,, 4, with the prescribed boundary conditions and extrapolated values V (0) m.. For k =,, s, we do the following calculations. For each ghost point (x i, y j ), use the inverse Lax-Wendroff type procedure to write the k-th order derivatives of g m (t) as a linear combination of k-th order normal derivatives plus tangential derivatives and other terms. Together with the extrapolation equations, form a linear system (such as (.35)) or a linear least squares problem for Û m (k), m =,, 4. Solve for Û m (k), m =,, Impose the value of Û i,j by the Taylor expansion (.33) and transform Û i,j to U i,j. We remark that the change of coordinate system is used only for determining inflow boundary conditions and for applying the inverse Lax-Wendroff procedure. Our uniform Cartesian mesh remains unchanged..6 Two-dimensional extrapolation We assume u i,j, (x i, y j ) Ω, are given. We aim to first construct a stencil E Ω for extrapolation and then obtain a (s k)-th order approximation of k u ˆx k (x,y)=(x0,y 0 ), which is

23 denoted by u (k), k = 0,, s. We assume s = 3, since we use this order of extrapolation in most of the numerical examples. The choice of stencil E is sketched in Figure.. E contains three one dimensional substencils S l, l = 0,,. Suppose the normal n (or ˆx-axis) intersects the grid line y = y n+l, l = 0,,, at a point P l. We identify the grid point on y = y n+l which is nearest to P l by (x l m, y n+l). Then we set S l = {(x l m, y n+l), (x l m, y n+l), (x l m+, y n+l)}, l = 0,, and E = l=0 S l. Notice that we might need to shift S l to the left or to the right so that S l lies in Ω. n ( x m, y n+ ) ( x m 0, y n ) P P 0 ( x m, y n+ ) P y n+ y n+ y n P 0 Figure.: The choice of stencil E (square points) for two-dimensional extrapolation Once the stencil E is chosen, we can easily construct a Lagrange polynomial in Q p (x, y) = a lm x l y m m=0 l=0 satisfying p (x i, y j ) = u i,j, (x i, y j ) E. (.36) u (k) is then obtained by u (k) = k ˆx p (x, y) k (x,y)=(x0,y 0 ). (.37) 3

24 We can also develop WENO type extrapolation, which is useful if a shock is close to the boundary. Suppose p r (x, y) = r m=0 r a lm x l y m, r = 0,,, l=0 is constructed on S r, where S 0 S S = E are two-dimensional substencils containing (r + ) points. We seek WENO type extrapolation of the form u (k) = r=0 ω r k p r (x, y) ˆx k (x,y)=(x0,y 0 ) where ω r are nonlinear weights. The nonlinear weights ω r are chosen to be, (.38) ω r = α r s=0 α s, with α r = d r (ɛ + β r ), where ɛ = 0 6, d 0 = x + y, d = x + y, d = d 0 d. β r are determined by β 0 = x + y β r = K α (D α p r (x, y)) dxdy, r =,, α r K where α is a multi-index and K = [x 0 x/, x 0 + x/] [y 0 y/, y 0 + y/]. We can show that β r, r = or, is small if a discontinuity appears in Sr and ω 0 = O( x + y ), ω = O( x + y ), (.39) ω = ω 0 ω, if the values of u i,j are smooth. We omit the proof here since the arguments are very similar to those in Section.. In the case of subsonic outflow boundary conditions, the high order Lagrange extrapolation (.37) seems to make the whole scheme mildly unstable. A remedy is to include more 4

25 points in E so that (.36) is satisfied in the sense of least squares. We use this least squares extrapolation in the vortex evolution problem in Section 3., where E is taken as { } E = (x µ, y ν ) Ω : (x µ x 0 ) + (y ν y 0 ) < 5 x. It seems that the WENO type extrapolation (.38) does not have this issue when used in the presence of shocks near the boundary. This might be due to the fact that the WENO type extrapolation reduces to a low order method if the solution becomes unstable, which helps stabilizing the scheme. 3 Numerical examples 3. One-dimensional examples Example We start with the wave equation u t + u x = 0 x (, ), t > 0, u(x, 0) = sin(πx) x [, ], u(, t) = g(t) t > 0. (3.) The left boundary x = is an inflow boundary, where a boundary condition is prescribed. The right boundary x = is an outflow boundary, where no boundary condition is needed. We first take g(t) = sin [π( + t)] (3.) so that the initial boundary value problem has a smooth exact solution u(x, t) = sin [π(x t)]. We use a fourth order Taylor expansion. The errors at t = are listed in Table 3.. We can clearly see the desired fifth order convergence. Next, we take g(t) as g(t) = { 0.5 t, t >. 5 (3.3)

26 Table 3.: Errors of the wave equation (3.) with boundary condition (3.). x = /N and t =. N L error order L error order 40.74E-05.3E E E E E E E E E- 5.5 The exact solution is then u(x, t) = x < t, 0.5 t x < t, sin [π(x t)] x t. For t, the exact solution has a discontinuity in its first derivative, due to the definition of g(t). For t >, a discontinuity enters the computational domain from the inflow boundary. We can observe from Figure 3. that both types of discontinuities are well captured by our method u(x) x Figure 3.: Wave equation (3.) with boundary condition (3.3). x = /40, t =.5. Solid line: exact solution; Symbols: numerical solution. 6

27 Example We next test the Burgers equation u t + ( u) = 0 x (, ), t > 0, x u(x, 0) = sin(πx) x [, ], u(, t) = g(t) t > 0. (3.4) Here g(t) = w(, t), where w(x, t) is the exact solution of the initial value problem on (, ) with periodic boundary conditions. For all t, the left boundary x = is an inflow boundary and the right boundary x = is an outflow boundary. We use a fourth order Taylor expansion. At t = 0.3, we have a smooth solution. The errors are listed in Table 3.. We achieve the designed fifth order accuracy. At t =., a shock is fully developed in the interior of the computational domain. A shock enters the inflow boundary at t = 8 and moves to x = 0 at t =. We can see from Figure 3. that the shock is well captured in both scenarios by our method. Table 3.: Errors of the Burgers equation (3.4). x = /N and t = 0.3. N L error order L error order 40 9.E E E E E E E E E E E E Example 3 For one-dimensional systems, we first test the Euler equations with smooth solutions. (a) The domain is ( π, π). The initial condition is ρ(x, 0) = + 0. sin x, u(x, 0) =, (3.5) p(x, 0) =. We want to impose the boundary conditions in such a way that the exact solution is simply 7

28 u(x) 0. u(x) x x (a) t =. (b) t = Figure 3.: Burgers equation (3.4), x = /40. Solid line: exact solution; Symbols: numerical solution. a translation of the initial condition ρ(x, t) = + 0. sin(x t), u(x, t) =, p(x, t) =. At both boundaries, we have λ < 0, λ > 0 and λ 3 > 0. Hence two boundary conditions are needed at x = π, which are taken as ρ( π, t) = + 0. sin t, (3.6) u( π, t) =. (3.7) One boundary condition is needed at x = π, which is taken as ρ(π, t) = + 0. sin t. (3.8) We use a fourth order Taylor expansion. We can observe from Table 3.3 that the designed fifth order accuracy is achieved. 8

29 Table 3.3: Density errors of the Euler equations with initial condition (3.5) and boundary conditions (3.6), (3.7), (3.8). x = π/n and t =. N L error order L error order E E E E E E E E E E (b) The domain is (0, ). The initial condition is ρ(x, 0) = a + sin(πx), u(x, 0) = + 0. sin(πx), (3.9) p(x, 0) = b, where a and b are real numbers. A reference solution W (x, t) can be obtained by the fifth order WENO scheme with periodic boundary conditions on (0, ) using an extremely refined mesh. To test our method, we assume Dirichlet boundary conditions in such a way that the exact solution of the initial boundary value problem is W (x, t). If we take a =.5, b = 3 in (3.9), we have λ < 0, λ > 0 and λ 3 > 0 at both boundaries. At the left boundary x = 0, we prescribe two boundary conditions u (0, t) = W (0, t), (3.0) u (0, t) = W (0, t). (3.) At the right boundary, we prescribe one boundary condition u (, t) = W (, t). (3.) We use a second order Taylor expansion so that a third order method is expected. For this truly nonlinear problem, we can see third order convergence in the left part of Table 3.4. If we take a =, b =.6 in (3.9), the eigenvalues λ m change signs at both boundaries. As a result, the number of boundary conditions varies with time. A least squares problem 9

30 Table 3.4: Density errors of the Euler equations with initial condition (3.9). x = /N. N a =.5, b = 3, t =.5 a =, b =.6, t = L error order L error order L error order L error order E-05.73E E-06.9E E E E E E E E E E E E E E E E E is solved if λ m is close to zero. We can observe from the right part of Table 3.4 that the designed third order accuracy is again achieved. Example 4 We then test our method for the Euler equations with shocks. We consider the interaction of two blast waves [9]. The initial data are U L 0 < x < 0., U(x, 0) = U M 0. < x < 0.9, U R 0.9 < x <, where ρ L = ρ M = ρ R =, u L = u M = u R = 0, p L = 0 3, p M = 0, p R = 0. There are solid wall boundary conditions at both x = 0 and x =. This problem involves multiple reflections of shocks and rarefactions off the walls. There are also multiple interactions of shocks and rarefactions with each other and with contact discontinuities. At both boundaries, we use a second order Taylor expansion with the inverse Lax-Wendroff procedure for the inflow condition u = 0 and the WENO type extrapolation for the outflow conditions. The density profile at t = is shown in Figure 3.3(a) with x = /800 and in Figure 3.3(b) with x = /600. The reference solution is computed by the fifth order WENO scheme with x = /6000, together with the reflection technique at both boundaries. We can clearly see that our boundary treatment gives a satisfactory resolution. 30

31 density 4 3 density x x (a) x = /800 (b) x = /600 Figure 3.3: The density profiles of the blast wave problem. Solid lines: reference solution computed by the fifth order WENO scheme with x = /6000, together with the reflection technique at boundaries; Symbols: numerical solutions by our boundary treatment. 3. Two-dimensional examples Example 5 We start our two-dimensional examples with the wave equation on a square or on a disk where u t + u x + u y = 0 (x, y) Ω, t > 0, u(x, y, 0) = sin [π(x + y)] (x, y) Ω, u(x, y, t) = g(x, y, t) (x, y) Γ, t > 0, (3.3) Ω = (, ) (, ), Γ = {(x, y) : x = or y = }, or Ω = {(x, y) : x + y < 0.5}, Γ = {(x, y) : x + y = 0.5 and x + y 0}. Both domains are illustrated in Figure 3.4 with a coarse mesh. Notice that the grid points are not located on the boundary in both cases. 3

32 (a) square domain (b) circular domain Figure 3.4: Domain Ω of the D wave equation (3.3). Square points indicate some of the grid points. Solid lines: inflow boundary; Dashed lines: outflow boundary. On the disk, special care must be taken when we impose the inflow boundary condition on a ghost point near the intersection of the inflow boundary and the outflow boundary, which is ( 0.5, 0.5) or (0.5, 0.5) in our example. In this situation, the inverse Lax-Wendroff procedure involves a small number divided by a small number, which ruins the accuracy or makes the scheme blow up. There are two ways to deal with this ill-conditioned problem. The first one is using the analytical expressions of time derivatives and tangential derivatives. The other one is adding an extrapolation equation and solving a least squares problem. We use the former here because the analytical expressions are available. We will use the latter in the next example. We take g(x, y, t) = sin [π(x + y t)] (3.4) so that we have a smooth exact solution u(x, y, t) = sin [π(x + y t)]. We use a fourth order Taylor expansion. The errors are listed in Table 3.5. We can clearly 3

33 Table 3.5: Errors of the D wave equation (3.3) with boundary condition (3.4). x = /N x, y = /N y. N x = N y on a square, t = 0.5 on a disk, t = 0.8 L error order L error order L error order L error order E-06.7E E E E E E E E E E E E E E E see fifth order convergence. We next take a discontinuous boundary condition g(x, y, t) = { sin π(x + y t) x + y t >.3, sin π(x + y t) x + y t.3. (3.5) Now we have a discontinuous exact solution. The numerical solution and exact solution along the diagonal are shown in Figure 3.5. We can see an excellent non-oscillatory resolution u(x) 0.8 u(x) x x (a) on a square, t = 0.5 (b) on a disk, t = 0.8 Figure 3.5: D wave equation (3.3) with boundary condition (3.5). x = y = /40. Cut along the diagonal. Solid line: exact solution; Symbols: numerical solution. 33

34 Example 6 We next test the D Burgers equation u t + (u ) x + (u ) y = 0 (x, y) Ω, t > 0, u(x, y, 0) = sin [π(x + y)] (x, y) Ω, u(x, y, t) = g(x, y, t) (x, y) Γ, t > 0, (3.6) where Ω = (, ) (, ), Γ = {(x, y) : x = or y = }, or Ω = {(x, y) : x + y < 0.5}, Γ = {(x, y) : x + y = 0.5 and x + y 0}. Here g(x, y, t) = w(x, y, t), where w(x, y, t) is the exact solution of the initial value problem on (, ) (, ) with periodic boundary conditions. We use a second order Taylor expansion at the inflow boundary. At the outflow boundary, we use the WENO type extrapolation, which is third order accurate if the solution is smooth. At t = 0.5, we have a smooth solution. The errors are listed in Table 3.6. We achieve the designed third order accuracy. At t = 0.55, a shock is fully developed in the interior of Ω. A shock begins entering Ω from the inflow boundary at t = 4 and moves to x = 0 at t = 6. We can see from Figure 3.6 that the shock is well captured on both domains. Notice that there is a shock very close to the outflow boundary in Figure 3.6(a). The robust WENO type extrapolation gives us a non-oscillatory numerical solution with little overshoot. Example 7 For two-dimensional systems, we test the vortex evolution problem for the Euler equations (.30). The mean flow is ρ = p = u = v =. We add to this mean flow an isentropic vortex perturbation centered at (x 0, y 0 ) in (u, v) and in the temperature T = p/ρ, no perturbation in the entropy S = p/ρ γ (δu, δv) = ɛ π e0.5( r) ( ȳ, x), 34

35 u(x) 0.8 u(x) x x (a) on a square, t = 0.55 (b) on a square, t = u(x) 0.8 u(x) x x (c) on a disk, t = 0.55 (d) on a disk, t = 6 Figure 3.6: D Burgers equation (3.6). x = y = /40. Cut along the diagonal. Solid line: exact solution; Symbols: numerical solution. 35

36 Table 3.6: Errors of the D Burgers equation (3.6). x = /N x, y = /N y, t = 0.5. N x = N y on a square on a disk L error order L error order L error order L error order 40.55E E-03.0E-04.77E E E E E E E E E E E E E E E E E (γ )ɛ δt = e ( r), 8γπ δs = 0, where ( x, ȳ) = (x x 0, y y 0 ), r = x + ȳ and the vortex strength is ɛ = 5. We regard the exact solution of this problem W (x, y, t) as the passive convection of the vortex with the mean velocity and take the boundary conditions from W (x, y, t) whenever needed. The number of boundary conditions is determined by the signs of the four eigenvalues λ m which vary both in space and in time. We use a second order Taylor expansion. In this D problem, least squares extrapolation is used instead of Lagrange extrapolation due to the stability issues mentioned in Section.6. We take the domain as a square Ω = ( 0.5, ) ( 0.5, ) or a disk Ω = {(x, y) : x +y < 0.5}. The density errors are listed in Table 3.7 and Table 3.8. The designed third order accuracy is achieved for this fully nonlinear problem. Table 3.7: Density errors of the vortex evolution problem on a square. The vortex is initially positioned at (0, 0). x =.5/N and t =. N x N y L error order L error order E-06.8E E E E E E E Example 8 We are most interested in applying our method to the solid wall boundary conditions (u, v) n = 0, when the wall is not aligned with the grid and can be curved. 36

37 Table 3.8: Density errors of the vortex evolution problem on a disk. The vortex is initially positioned at (0.3, 0.3). x = /N and t = 0.. N x N y L error order L error order E-07.9E E E E E E E Our first example of this kind is the double Mach reflection problem [9]. This problem is initialized by sending a horizontally moving shock into a wedge inclined by a 30 angle. In order to impose the solid wall condition by the reflection technique, people usually solve an equivalent problem that puts the solid wall horizontal and puts the shock 60 angle inclined to the wall, see for example [8] and [5]. Another way to avoid the trouble of imposing boundary conditions is to use a multidomain WENO method [4]. With the use of our method, we are able to solve the original problem with a uniform mesh in a single domain. The computational domain is shown in Figure 3.7(a), together with some of the grid points near the wall which indicate that the wall is not aligned with the grid. Initially a right-moving Mach 0 shock is positioned at (0,0) making an angle of 90 with the x-axis. At y = 0, the exact postshock condition is imposed. At the top boundary y = 3 + 3, the flow values are set to describe the exact motion of the Mach 0 shock. At the wall, we use a second order Taylor expansion with the inverse Lax-Wendroff procedure and the WENO type extrapolation. Figure 3.7(b) shows the density contour on a mesh with x = y =. A zoomed-in region near the double Mach stem is shown in 30 Figure 3.8(a). We rotate and translate the region for ease of comparison. In Figure 3.8(b), we show the result of the regular fifth order WENO scheme for the equivalent problem with a regular reflective boundary condition on a mesh with a comparable size. Figure 3.8(c) and Figure 3.8(d) show the density contours on a refined mesh. We can see that the results of our boundary treatment are very similar to those obtained by the reflection technique. The slight difference comes perhaps from the fact we impose the no-penetration condition 37

38 strongly while the reflection technique imposes it weakly. ( 3 ), ( 3 ) 3 3, Y wall ( 3 ) 3 3, 0.5 ( ) 3, 0 ( 0, 0) X (a) computational domain (b) density contour Figure 3.7: Left: The computational domain (solid line) of the double Mach reflection problem. The dashed line indicates the computational domain used in [8] and [5]. The square points indicate some of the grid points near the wall. Illustrative graph, not to scale; Right: Density contour of double Mach reflection, 30 contours from.73 to 0.9. x = y =. 30 Example 9 Our final example involves a curved wall which is a circular cylinder of unit radius positioned at the origin on a x-y plane. The problem is initialized by a Mach 3 flow moving toward the cylinder from the left. In order to impose the solid wall boundary condition at the surface of the cylinder by the reflection technique, a particular mapping from the unit square to the physical domain is used in [8]. Using our method, we are able to solve this problem directly in the physical domain, which is shown in Figure 3.9, together with some of the grid points near the cylinder which indicate the wall cuts the grid in an arbitrary fashion. Our computational domain is the upper half of the physical domain, due to the symmetry of this problem. At y = 0, the reflection technique is used. At the left inflow boundary x = 3, the uniform far-field data is imposed. At the top boundary y = 6 and the right boundary x = 0, constant extrapolation is used because of the hyperbolic nature of this problem. The inverse Lax-Wendroff procedure and the WENO type extrapolation 38