A high order moving boundary treatment for compressible inviscid flows 1. Abstract

Transcription

1 A high order moving boundary treatment for compressible inviscid flows Sirui Tan and Chi-Wang Shu Abstract We develop a high order numerical boundary condition for compressible inviscid flows involving complex moving geometries. It is based on finite difference methods on fixed Cartesian meshes which pose a challenge that the moving boundaries intersect the grid lines in an arbitrary fashion. Our method is an extension of the so-called inverse Lax-Wendroff procedure proposed in [6] for conservation laws in static geometries. This procedure helps us obtain normal spatial derivatives at inflow boundaries from Lagrangian time derivatives and tangential derivatives by repeated use of the Euler equations. Together with high order extrapolation at outflow boundaries, we can impose accurate values of ghost points near the boundaries by a Taylor expansion. To maintain high order accuracy in time, we need some special time matching technique at the two intermediate Runge-Kutta stages. Numerical examples in one and two dimensions show that our boundary treatment is high order accurate for problems with smooth solutions. Our method also performs well for problems involving interactions between shocks and moving rigid bodies. AMS subject classification: 65M06 Keywords: numerical boundary conditions; complex moving boundaries; inverse Lax- Wendroff procedure; compressible inviscid flows; no-penetration conditions Research supported by AFOSR grant FA and NSF grant DMS Division of Applied Mathematics, Brown University, Providence, RI sirui@dam.brown.edu Division of Applied Mathematics, Brown University, Providence, RI shu@dam.brown.edu

2 Introduction In this paper, we develop a high order accurate numerical boundary condition based on finite difference methods for simulations of compressible inviscid flows involving complex moving geometries. For problems in such geometries, it is difficult to use body-fitted meshes which conform to the moving geometry. Instead, methods based on fixed Cartesian meshes have been successfully developed. For example, the immersed boundary (IB) method introduced by Peskin [] is widely used to solve incompressible flows in complicated time-varying geometries. See also Mittal et. al [] for an overview of the method and its applications. The IB method is extended for compressible viscous flows by De Palma et al. [4], Ghias et al. [8] and de Tullio et al. [5]. One of the challenges of the IB method is the representation of the moving objects which cut through the grid lines in an arbitrary fashion. To solve compressible inviscid flows in complex moving geometries, most methods in the literature are based on finite volume schemes [, 7, 9, 5]. The challenge mainly comes from the so-called small-cell problem. Namely, one obtains irregular cut cells near the boundary, which may be orders of magnitude smaller than the regular grid cells, leading to a severe time step restriction. Forrer et al. [7] fill in the cut cells and a set of ghost cells by a mirror flow extrapolation over the reflecting wall. Arienti et al. [] develop an Eulerian-Lagrangian coupling scheme. Hu et al. [9] develop a conservative interface method based on the level set technique suitable not only for complex boundary problems but also for multi-fluid problems. Small cut cells are treated with a mixing procedure. Shyue [5] proposes a moving-boundary tracking algorithm based on finite volume wave-propagation methods. The method is stable under the standard CFL condition relative to the regular cell size, even if there are very small cut cells near the tracked interface. In terms of accuracy, all the finite volume schemes mentioned above are at most second order. In particular, the errors at the boundaries sometimes fall short of second order, see numerical examples in [, 9]. Our high order finite difference scheme on fixed Cartesian meshes is based on the so-

3 called inverse Lax-Wendroff (IL-W) procedure proposed in our previous paper [6], where boundary treatment for conservation laws in static geometries is considered. At the inflow boundaries, we repeatedly use the partial differential equations (PDEs) 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 impose accurate values of ghost points near the boundary by a Taylor expansion. At outflow boundaries, various extrapolation of high order accuracy is used. For Euler equations, extrapolation is written in the form of Taylor expansions so that inflow and outflow boundary conditions are coupled by a linear system of equations with normal derivatives as unknowns. Numerical examples in [6] show that this approach is high order accurate and the time step is not limited by the numerical boundary conditions. If we try to extend this method to fluid flow problems with moving boundaries, there are two underlying issues. First, the no-penetration condition is prescribed in the Lagrangian specification. As a result, we should use material derivatives instead of Eulerian time derivatives in the IL-W procedure. Second, in expansion flows there may be grid points which are outside the computational domain in the previous time level t n but enter the computational domain in the current time level t n. We call such grid points newly emerging points. Their values can be imposed by the same way as the values of ghost points. As in [7, 9, 5], we treat rigid bodies moving at a prescribed motion or in interaction with the fluid. The problem is described in detail in Section. In Section, we develop our high order boundary treatment based on the IL-W procedure for one-dimensional (D) problems. We shall see special care must be taken to maintain third order accuracy when we construct values of ghost points in the two intermediate Runge-Kutta stages of the third order Runge- Kutta method. The method is then generalized to two-dimensional (D) problems with general geometries. Numerical examples are presented in Section 4 to demonstrate the high order accuracy and robustness of our approach. Concluding remarks are given in the last section.

4 Problem description The compressible Euler equations in D are given as U t + F (U) x + G(U) y = 0, x = (x, y) Ω(t), 0 < t < T, (.) where U = ρ ρu ρv E, F (U) = ρu ρu + p ρuv u(e + p), G(U) = ρv ρuv ρv + p v(e + p) ρ, u = (u, v) T, p and E describe the density, 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. To describe the boundary conditions, we denote the boundary of Ω(t) by Γ(t) and the normal vector at a point x on Γ(t) by n(x, t). The sign of n(x, t) is chosen in such a way that it is positive if it points to the fluid from the rigid body. X b (a, t) represents the position vector (in Eulerian coordinates) of a point a on Γ(t), where a is the Lagrangian coordinate of the point. We mainly consider rigid bodies moving at a prescribed motion. Namely, X b (a, t) and thus Ω(t) are explicitly given. The no-penetration boundary condition for inviscid flows is then where V b (t) = X b t u(x, t) n(x, t) = V b (t) n(x, t), for all x = X b (a, t) Γ(t), (.) is the prescribed velocity. Notice that V b (t) is independent of a for rigid bodies. If the motion of a rigid body is induced by the fluid, the acceleration can be expressed as X b = pnds, (.) t M b Γ(t) where M b is the rigid body mass. Although X b (a, t) is not explicitly given in this case, we can obtain it at each time level by integrating (.) in time. 4

5 Scheme formulation We assume the domain Ω(t) is covered by a fixed Cartesian mesh with mesh size x = y = h for all t [0, T ]. The semi-discrete approximation of (.) is given by d dt U ij(t) = ( ˆF i+/,j h ˆF ) i /,j ) (Ĝi,j+/ h Ĝi,j /, (.) where ˆF i+/,j and Ĝi,j+/ are numerical fluxes. We use a third order total variation diminishing Runge-Kutta (R-K) method [4] to integrate the system of ordinary differential equations (.) in time U () i,j = U n i,j + tl ( U n i,j ), U () i,j = 4 U n i,j + 4 U () i,j + ( 4 tl U n+ i,j = U n i,j + U () i,j + tl ( U () i,j U () i,j ), (.) ), where L ( ) is the operator defined by the right-hand side of (.). In addition to the standard CFL conditions determined by the interior schemes, we assume Γ(t) travels a distance of at most h in each direction from t n to t n+, i.e., t < h max t [tn,t n+ ] V b (t). (.) For definiteness, we use the fifth order finite difference weighted essentially non-oscillatory (WENO) scheme with Lax-Friedrichs flux splitting [0] to form the numerical fluxes, although our boundary treatment is independent of the interior scheme. The fifth order WENO scheme requires a seven point stencil in both x and y directions. Thus near Γ(t n ) where the numerical stencil is partially outside of Ω(t n ), up to three ghost points are needed in each direction. Notice that we should update values of all grid points inside Ω(t n ), which include values of newly emerging points, from time level t n to time level t n+. For newly emerging points, we not only need to construct their values at time level t n, but also need one extra ghost point in each direction to update their values because of (.). See Fig.. for a demonstration in both D and D. If the same method is used to construct values of ghost points and newly 5

6 emerging points, there is actually no need to distinguish them in implementation. We only need to construct values of four ghost points in each direction. We concentrate on this issue in the rest of the paper.. One-dimensional problems We assume the left boundary is a moving wall with prescribed position X b (t) and velocity V b (t) = X b (t). Since the boundary condition (.) is prescribed in velocity u, we rewrite the Euler equations in terms of primitive variables where W = W W W W t + A(W )W x = 0, (.4) = ρ u p, A(W ) = u ρ 0 0 u /ρ 0 ρc u and c = γp/ρ is the sound speed. At time level t = t n, we have grid points x < x < x X b (t n ) < x 0 < x < < x N. We assume x,, x 0 are the four ghost points in which x 0 is a newly emerging point, i.e., X b (t n ) < x 0 < X b (t n ), see Fig..(a). U,, U N, or W,, W N, have been updated from time level t n to time level t n. Here we suppress the t n dependence without causing any confusion. We proceed as in [6] to construct values of ghost points W,, W 0 by a (s )th order Taylor expansion, (W m ) j = s k=0 (x j X b (t n )) k k! Wm (k), m =,,, j =,, 0, (.5) where W (k) m is a (s k)th order approximation of the spatial derivative k W m x k x=xb (t n), t=t n. We first do a local characteristic decomposition of the PDEs at the boundary x = X b (t n ). We assume A(W ) has three eigenvalues λ = u c, λ = u, λ = u + c and a complete set of left eigenvectors l m (W ), m =,,, which forms a matrix l (W ) l, l, l, L(W ) = l (W ) = l, l, l,. l (W ) l, l, l, 6

7 (a) D (b) D Figure.: Newly emerging points and ghost points at time level t n. 7

8 The two outgoing local characteristic variables V and V at grid points near the boundary are defined by (V m ) j = l m (W )W j, m =,, j =,, s. (.6) We extrapolate (V m ) j, m =,, to x = X b (t n ) either with Lagrange extrapolation if the solution is smooth near the boundary or with the so-called WENO type extrapolation if a shock is close to the boundary. For both types of extrapolation, see (.) and (.) in [6]. The extrapolated kth order derivative of V m at the boundary is denoted by V (k) m, k = 0,, s. W (0) can be naturally imposed by W (0) = V b (t n ) = X b (t n) according to (.). The other two components, W (0) and W (0), should be obtained by extrapolated values V (0) m. We have a linear system with Wm (0) 0 0 l, l, l, l, l, l, W (0) W (0) W (0) as unknowns = X b (t n) V (0) V (0). (.7) Next we try to find the spatial derivatives W () m with the IL-W procedure for u, together with the extrapolation of V m, m =,. The second equation of (.4) can be written in Lagrangian form as where D Dt = t + u x Du Dt + ρ p x = 0, is the material derivative. Thus the material derivative of u, which is actually X b (t n), can be converted to the spatial derivative p x at the boundary. We then have a linear system with Wm () 0 0 W (0) l, l, l, l, l, l, as unknowns W () W () W () = X b (t n) V () V (). (.8) Repeatedly using the Euler equations (.4), we are able to convert higher order material derivatives to spatial derivatives, and thus to form a linear system with the spatial derivatives W (k) m as unknowns and the right-hand vector depending on the derivatives of X b (t), 8

9 extrapolated derivatives of V, V and lower order spatial derivatives W (l) m, l = 0,, k. For example, the linear system in the case of k = is (0) 0 γw 0 W () l, l, l, W () = l, l, l, W () W (0) X b (t n) γw () W () V () V (). (.9) Notice that the numerical method we have described so far is for time level t n only. We need to match the time levels when constructing values of ghost points in the two intermediate stages U () j and U () j of the R-K method (.). The traditional match of time U () j t n+, U () j t n + t/ decreases the accuracy to second order []. One idea is to update the position X b and the velocity V b at each R-K stage by X () b = X b (t n ) + tx b (t n), V () b = X b (t n) + tx b (t n); (.0) The updated positions X (i) b X () b = 4 X b(t n ) + 4 X() b + () tv b 4 = X b (t n ) + tx b(t n ) + 4 t X b (t n ), V () b = X b(t n ) + tx b (t n ) + 4 t X b (t n ). (.) and velocities V (i) b, i =,, instead of X b (t n ) and X b (t n), are used in the two intermediate stages. This time matching technique successfully maintains third order accuracy in Lagrangian type schemes []. However, Example in Section 4 shows that it is only second order accurate if applied here. The reason is probably that our mesh does not move with the fluid. As in [, 6], we can achieve third order accuracy by the following match of time u () X b (t n) + t u t u () X b(t n ) + t u t, (.) x=xb (t n), t=t n + u x=xb (t n), t=t n 4 t. (.) t x=xb (t n), t=t n 9

10 The Eulerian time derivatives can be obtained by a standard Lax-Wendroff procedure, since all the necessary spatial derivatives have been obtained at time level t n. For example, where u = u u t x p ρ x, u = u u t t x u u x t + ρ ρ t p x ρ p x t, (.4) u x t ρ t p x t ( ) u = u u x x + ρ p ρ x x ρ = u ρ x ρ u x, = (γ + ) p x u x u γp x u p x. p x, Our IL-W procedure should also be adjusted in the two intermediate R-K stages. We write spatial derivatives in terms of Eulerian time derivatives instead of material derivatives. For example, the first equation of (.8) is replaced by W (0) W () = u t + t u (.5) x=xb (t n), t=t n t x=xb (t n), t=t n in the first stage; replaced by W (0) W () in the second stage. W () W (0) W () W (0) = u t + t x=xb (t n), t=t n u t (.6) x=xb (t n), t=t n Here we summarize our algorithm for s =. It is a third order method that we implement for the numerical examples in Section 4. We assume U,, U N have been updated from time level t n to time level t n. Our goal is to impose values of ghost points W,, W 0 at time level t n and the two intermediate R-K stages. For time level t n :. Do a local characteristic decomposition of the Euler equations (.4). Form the two outgoing characteristic variables (V m ) j, m =,, j =,,, as in (.6). Extrapolate 0

11 (V m ) j to the boundary to obtain Vm (k), k = 0,,, with Lagrange extrapolation or the WENO type extrapolation (see (.) and (.) in [6]). Solve for W (0) m, m =,,, in (.7).. Use the IL-W procedure with material derivatives and the extrapolation equations to form linear system (.8). Solve for W () m, m =,,.. Form linear system (.9). Solve for W () m, m =,,. 4. Use the standard Lax-Wendroff procedure (.4) to compute u and u t t at the boundary. 5. Impose the values of the ghost points by the Taylor expansion (.5). For the first and second intermediate R-K stages: 6. Do Step but replace X b (t n) in (.7) by the right-hand side of (.) and (.) respectively. 7. Form linear system (.8) but replace the first equation by (.5) and (.6) respectively. Solve for W () m, m =,,. 8. Do Step and then do Step 5. Notice that Step is shared by all the R-K stages because only a first order approximation of the second spatial derivatives is needed for a third order scheme.. Two-dimensional problems We now turn to the D problem (.) with the no-penetration boundary condition (.) in which X b (a, t) and V b (t) are prescribed. We assume the values of all the grid points inside Ω(t n ) have been updated by the interior scheme from time level t n to time level t n. At time level t n, remember that in each direction we need to impose values of four ghost points, one of which is possibly a newly emerging point, see Fig..(b). For a ghost point

12 P = (x i, y j ), we find a point P 0 = (x 0, y 0 ) = x 0 on the boundary Γ(t n ) so that the normal n(x 0, t n ) 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 y ) = T ( x y ), (.7) where θ is the angle between the normal n(x 0, t n ) and the x-axis and T is a rotational matrix. The ˆx-axis then points in the normal direction to Γ(t n ) at P 0 and the ŷ-axis points in the tangential direction to Γ(t n ) at P 0, see Fig... In this local coordinate system, the Euler equations (.) are written in terms of primitive variables as Ŵ t + A(Ŵ ) Ŵ ˆx + B(Ŵ ) Ŵ ŷ = 0, (.8) where Ŵ = Ŵ Ŵ Ŵ Ŵ 4 = ρ û ˆv p, A(Ŵ ) = û ρ û 0 ρ 0 0 û 0 0 ρc 0 û, B(Ŵ ) = ˆv ρ ˆv ˆv ρ 0 0 ρc ˆv and (û, ˆv) T = T (u, v) T. Our IL-W procedure is carried out by the use of (.8), i.e., in the local coordinate system (.7)., Figure.: The local coordinate system (.7).

13 For a third order boundary treatment, the value of the ghost point (x i, y j ) is imposed by the Taylor expansion (Ŵm) i,j = d k k! Ŵ m (k), m =,, 4, (.9) k=0 where d is the ˆx-coordinate of P and Ŵ (k) m is a ( k)th order approximation of the normal derivative kŵm ˆx k (x,y)=x0, t=t n. We assume Ŵ 0 is the value of a grid point nearest to P 0 among all the grid points inside Ω(t n ). A(Ŵ 0) has four eigenvalues λ = û 0 c 0, λ = λ = û 0, λ 4 = û 0 + c 0 and a complete set of left eigenvectors l m (Ŵ 0), m =,, 4, which forms a matrix L(Ŵ 0) = l (Ŵ 0) l (Ŵ 0) l (Ŵ 0) l 4 (Ŵ 0) = l, l, l, l,4 l, l, l, l,4 l, l, l, l,4 l 4, l 4, l 4, l 4,4 The three outgoing characteristic variables V m, m =,,, at grid points near P 0 can be defined by. (V m ) µ,ν = l m (Ŵ 0)Ŵ µ,ν, m =,,, (x µ, y ν ) E i,j, (.0) where E i,j Ω(t n ) is a set of grid points used to construct an extrapolating polynomial. We extrapolate (V m ) µ,ν to P 0 and denote the extrapolated kth order ˆx-derivative of V m by V (k) m, k = 0,,. For the choice of the extrapolation stencil E i,j and various D extrapolation, see Section.6 in [6]. We solve Ŵ m (0), m =,, 4, by a linear system of equations Ŵ (0) l, l, l, l,4 Ŵ (0) l, l, l, l,4 Ŵ (0) = l, l, l, l,4 Ŵ (0) 4 V b (t n ) n(x 0, t n ) V (0) V (0) V (0). (.) Here the first equation is nothing but the no-penetration boundary condition (.). The other equations represent extrapolation of outgoing characteristic variables. Next, we take the first material derivative D Dt = t + û ˆx + ˆv ŷ Dû Dt + û Dn Dt of (.) and obtain = d dt (V b n),

14 where û = (û, ˆv) T. Converting the material derivative Dû to spatial derivatives by the second Dt equation of (.8), we have [ p ˆx = ρ û Dn Dt d ] dt (V b n). The right-hand side of the above equation is already known if evaluated at P 0. As a result, Ŵm (), m =,, 4, can be solved by l, l, l, l,4 l, l, l, l,4 l, l, l, l,4 Ŵ () Ŵ () Ŵ () Ŵ () 4 = b, (.) where b = Ŵ (0) [ (Ŵ ) T (0), Ŵ (0) Dn d (V Dt dt b n)] (x,y)=p 0, t=t n V () V () V (). Taking the second material derivative of (.), we obtain D û Dt + Dû Dt Dn Dt + û D n Dt = d dt (V b n). In the above equation, we have already known all the terms evaluated at P 0 except the first one. By the IL-W procedure, we convert the material derivative D û Dt and obtain [ γp û d = ρ ˆx dt (V b n) Dû Dt Dn ] Dt û D n Dt γ û ˆx p p (γ ) ˆx ˆx ˆv ŷ ˆv ˆx p ˆv γp ŷ ˆx ŷ. to spatial derivatives (.) Notice that the last three terms in (.) contain tangential derivatives which can be computed by numerical differentiation since we have obtained Ŵ (0) m and Ŵ () m of all the ghost points. For robustness, numerical differentiation is done by a least squares polynomial of suitable degree if the solution is smooth near the boundary, or by WENO type differentiation 4

15 otherwise. We then form the linear system with Ŵ m () as unknowns by adding extrapolation equations of V, V and V to (.). We have finished describing our approach for the time level t n. For the two intermediate R-K stages, we follow the same time matching technique as for D problems. We now summarize our third order method for the boundary condition (.) in D with a prescribed boundary motion. We assume the values of all the grid points inside Ω(t n ) have been updated by the interior scheme from time level t n to time level t n. Our goal is to impose the value of (Ŵm) i,j, m =,, 4, for each ghost point (x i, y j ). For time level t n :. For each ghost point (x i, y j ), we do the following calculations. Decide the local coordinate system (.7). Do a local characteristic decomposition of the Euler equations (.8). Form the three outgoing characteristic variables (V m ) µ,ν, m =,,, (µ, ν) E i,j, as in (.0). Extrapolate (V m ) µ,ν to the boundary to obtain Vm (k), k = 0,,, with least squares extrapolation or the WENO type extrapolation (see Section.6 in [6]). Solve for Ŵ (0) m, m =,, 4, in (.).. For each ghost point (x i, y j ), use the IL-W procedure with material derivatives and the extrapolation equations to form linear system (.). Solve for Ŵ () m, m =,, 4.. For each ghost point (x i, y j ), form a linear system with Ŵ () m extrapolation equations of V, V and V. Solve for Ŵ () m, m =,, Use a standard Lax-Wendroff procedure to compute û for all the ghost points. t (x,y)=p0, t=t n 5. Impose the values of all the ghost points by the Taylor expansion (.9). For the first and second intermediate R-K stages: as unknowns by (.) and and û t (x,y)=p0, t=t n 6. For each ghost point (x i, y j ), do Step but replace V b (t n ) n(x 0, t n ) in (.) by V b (t n ) n(x 0, t n ) + t û t 5 (x,y)=p0, t=t n

16 and V b (t n ) n(x 0, t n ) + t û t + t (x,y)=p0, t=t n 4 û t (x,y)=p0, t=t n respectively. 7. For each ghost point (x i, y j ), form linear system (.) but replace the first equation by Ŵ (0) Ŵ () Ŵ () 4 = û t + t û + (x,y)=p0, t=t n t Ŵ (0) û ŷ (x,y)=p0, t=t n Ŵ (0) and Ŵ (0) Ŵ () Ŵ () 4 Ŵ (0) = û t + t (x,y)=p0, t=t n û t + Ŵ (0) û ŷ (x,y)=p0, t=t n respectively. Solve for Ŵ () m, m =,, Do Step and then do Step 5. If the motion of a rigid body is not prescribed but induced by the fluid, our algorithm should be adjusted as follows. Before Step, we compute dv b dt = X b t by (.), since we have obtained pressure p at P 0 on Γ(t n ) for all the ghost points. The integral in (.) can be calculated by the trapezoidal rule, for example. Before Step, we compute d V b dt the material derivative of (.), i.e., d V b dt = M b Γ(t) D (pn) ds. Dt by taking Notice that integrating (.) in time by the R-K method (.), we can obtain X b (a, t n+ ) and V b (t n+ ) so that our algorithm can be continued at next time level t n+. 4 Numerical examples In this section, we show some numerical examples to demonstrate that our method is third order accurate if the solution is smooth. Our method also performs well for problems involving interactions between shocks and moving rigid bodies. Throughout our numerical tests, 6

17 we use the third order boundary treatment (s = ). The CFL number is taken as 0.6, unless otherwise indicated. (.) is implied by the CFL conditions in all our examples. Example We consider a gas confined between two rigid walls. The right wall is fixed at x r =.0 while the left wall is moving. We assume the left wall is positioned at x l (t). The initial conditions are ρ(x, 0) = + 0. cos [π (x 0.5)], u(x, 0) = x, p(x, 0) = ρ(x, 0) γ, such that the initial entropy s(x, 0) =. As long as the solution stays smooth, we have isentropic flow, i.e., s(x, t) =. Thus the numerical value of the entropy can be used for the analysis of convergence. This example is considered by Forrer et al. [7] and later by Arienti et al. [] and Hu et al. [9]. Second order convergence in total entropy is reported in all three papers. However, the error of the entropy at the left moving wall falls short of second order in [, 9]. In our first case, we set x l (t) = 0.5( t) such that the wall moves with a constant speed. Our high order boundary treatment is used at the left moving boundary while the standard reflection technique is used at the right fixed boundary. We measure the L errors and L errors in entropy at t = 0.5. We can see from Table 4. that our method is third order accurate even near the moving boundary. Table 4.: Entropy errors and convergence rates of Example. x l (t) = 0.5( t). h L error order L error order /80 5.9E-08.5E-06 /60 5.8E E /0 5.50E-0.6.E / E-.0.99E-09.8 / E E-0.9 / E-.0.5E-.49 7

18 In our second case, we set x l (t) = 0.5( sin t). Now the speed of the wall varies in time. The left part of Table 4. shows the results of our method at t = 0.5 while the right part shows the results at t = 0.5 obtained by the same spatial discretization but with the time matching technique (.0) and (.). We can see that we still achieve the designed third order accuracy by our method. However, the boundary error of the other method decays to second order on refined meshes. Table 4.: Entropy errors and convergence rates of Example. x l (t) = 0.5( sin t). h our time matching technique technique (.0) and (.) L error order L error order L error order L error order /80 5.E-08.7E E E-07 / E-09.5.E E-09..5E /0 5.70E-0.0.0E E-0.0.0E / E-.6.7E E E-09.8 /80 7.4E-.0.54E E E-0.96 / E E-.9.E-.9.E-0.79 Example This is a D problem involving shocks and rarefaction waves. A piston with width 0h is initially centered at x = 5h inside a shock tube. Here h is the mesh size. The piston instantaneously moves with a constant velocity u p = into an initially quiescent fluid with ρ = and p = 5/7. This problem is equivalent to two independent Riemann problems and thus the exact solution can be obtained. A shock forms ahead of the piston and a rarefaction wave forms in the rear. A D version of this problem is considered by Murman et al. []. Shyue [5] tests a D moving piston problem with a different Mach number. We take h = 0.5, the same mesh size as in [], and set the CFL number to be 0.5. The density and pressure profiles at t = are plotted in Fig. 4., together with the exact solution. Our approach predicts the correct shock location on this relatively coarse mesh, indicating that the mass, momentum and energy loss through the piston is quite small. Moreover, our numerical density does not suffer from the undershoot just ahead of the piston which appears in Fig. 0(a) of [] and in Fig. of [5]. The numerical solution of the rarefaction wave also agrees well with the exact solution. 8

19 density pressure x x (a) Density (b) Pressure Figure 4.: Density and pressure profiles of Example. The piston is represented by the rectangle. Solid lines: exact solutions; Symbols: numerical solutions with h = 0.5. Example We now move on to D examples. We first test a D version of Example. A gas is confined in a rectangular region whose boundaries are rigid walls. The top and bottom walls are fixed at y = 0 and y = respectively. The right wall is fixed at x =. The left moving wall is positioned at x l (t) = 0.5( sin t). The initial conditions are ρ(x, y, 0) = + 0. cos [π (x 0.5)] + 0. cos [π(y 0.5)], u(x, y, 0) = x, v(x, y, 0) = y( y) cos(πx), p(x, y, 0) = ρ(x, y, 0) γ, such that the initial entropy s(x, y, 0) =. We use our high order boundary treatment at the left moving wall and the reflection technique at the fixed walls. The L errors and L errors in entropy at t = 0.5 are listed in Table 4.. We achieve the designed third order accuracy for this D problem. Example 4 Our next example involves D flows in complex moving geometries. We follow 9

20 Table 4.: Entropy errors and convergence rates of Example. h L error order L error order /80.50E-08.8E-07 /60.0E E-08.4 /0 9.70E E-09. / E E-0. the idea in Example to construct isentropic flows such that we are able to measure the entropy errors and to analyze the rate of convergence. The computational domain is [ 4, 4] [ 4, 4] with all the boundaries as rigid walls. A rigid cylinder with radius R = is initially centered at (0, 0) and starts moving. The center of the cylinder is positioned at X c (t). We use our high order boundary treatment at the surface of the moving cylinder and the reflection technique at the fixed walls. The initial conditions should be consistent with the no-penetration boundary condition (.). For this purpose, we define ũ(x, y) = λ (x, y)u (x, y) + λ (x, y), where and λ (x, y) = λ (x, y) = (4 )( x + y ) ( 6 + y )( 6 + x ), x + y (x 6) (y 6) ( ) ( ), x y 6 6 x +y x +y ( π ) ( π ) u (x, y) = sin 4 x sin 4 y. We also define ṽ (x, y) = λ (x, y)v (x, y) + λ (x, y)v (x, y), ṽ (x, y) = λ (x, y)v (x, y) + λ (x, y), where ( π ) ( π ) v (x, y) = sin 4 x sin 4 y, 0

21 v (x, y) = 6 Notice that ũ(x, y) satisfies the following conditions ( x + y ) sin ( π 4 x ). ũ(x, y) = 0 if x = ±4, ũ(x, y) = if x + y =, (4.) ũ y = 0 if y = ±4. Similar conditions hold for ṽ (x, y) and ṽ (x, y). The last equation of (4.) guarantees that the reflection technique gives us at least third order accuracy at the fixed horizontal walls so that we can concentrate on the treatment of the moving boundary. In our first case, we take X c = ( 0.5 sin t, 0) such that the cylinder moves horizontally. The initial conditions are ρ(x, y, 0) =, u(x, y, 0) = 0.5ũ(x, y), v(x, y, 0) = 0.5ṽ (x, y) and p(x, y, 0) =. We run the test to time t = 0.4 when the solution keeps smooth. Fig. 4.(a) shows the density contour plot at t = 0.4 with h = /40. We can see that the density ahead of the cylinder is generally larger than its initial value due to the compression. A shock will start to develop shortly afterwards. The left part of Table 4.4 lists the L errors and L errors in entropy, which give us third order convergence. Table 4.4: Entropy errors and convergence rates of Example 4. t = 0.4. h X c = ( 0.5 sin t, 0) X c = ( 0.5 sin t, 0.t) L error order L error order L error order L error order /5.8E-0 7.7E-0.95E-0 9.E-0 /0.80E-0..E E-0..4E-0.69 /0.9E E E E-04.5 /40.79E E E E-05.8 /80.95E E-06.5.E E-06.7 In the second case, we take X c = ( 0.5 sin t, 0.t) such that the cylinder moves in the D space. The initial conditions are ρ(x, y, 0) =, u(x, y, 0) = 0.5ũ(x, y), v(x, y, 0) = 0.ṽ (x, y) and p(x, y, 0) =. At t = 0.4, the density contour plot with h = /40 is shown in Fig. 4.(b) and the entropy errors are listed in the right part of Table 4.4. Third order convergence is again achieved.

22 Y Y X X 0.9 (a) X c = ( 0.5 sin t, 0) (b) X c = ( 0.5 sin t, 0.t) Figure 4.: Density contours of Example 4. h = /40, t = 0.4. Example 5 The last example shows that our high order method can also treat a rigid body whose motion is induced by the fluid. We test the so-called cylinder lift-off problem which is first proposed by Falcovitz et al. [6] and considered in [, 7, 9, 5] later. In this problem, a rigid cylinder initially resting on the floor of a D channel is driven and lifted by a strong shock. The problem setup is the same as in [, 7, 5]. The computational domain is [0, ] [0, 0.]. A rigid cylinder with radius 0.05 and density 0.77 is initially centered at (0.5, 0.05). A Mach shock starts at x = 0.08 moving towards the cylinder. The density and pressure of the resting gas are ρ =.4 and p =.0 respectively. The top and bottom of the domain are rigid walls. The left boundary is set to the post-shock state and the right boundary is supersonic outflow. We use our boundary treatment at the surface of the moving cylinder and the reflection technique at the top and bottom walls. Since the cylinder initially rests exactly on the floor, a stencil for high order extrapolation may be too wide to be contained in the computational domain. We have to use low order extrapolation in this situation and turn to high order

23 extrapolation otherwise. We list the center of the cylinder at two fixed times for different meshes in Table 4.5. The results imply a superlinear convergence rate. Similar rate is obtained in [9], whereas linear convergence rate in relative mass loss is reported in [, 5]. We plot pressure contours at t = 0.64 and t = in Fig. 4. and Fig. 4.4 respectively. The flow structures agree with those in [, 9, 5]. Table 4.5: Center of the cylinder of Example 5 h t = 0.64 t = x-coordinate y-coordinate x-coordinate y-coordinate / E E E-0.759E-0 /0.65E-0 8.9E E E-0 / E E E-0.457E-0 /80.559E-0 8.4E E E-0 / E E-0 6.6E-0.468E-0 5 Concluding remarks In this paper, we develop a high order numerical boundary condition for compressible inviscid flows involving complex moving geometries. It is based on finite difference methods on fixed Cartesian meshes which are efficient and easy to implement compared with body-fitted meshes for our problems. We follow the idea of the IL-W procedure proposed in [6] for static geometries and successfully extend it to fluid flow problems in moving geometries. Instead of converting Eulerian time derivatives to spatial derivatives as in [6], we use the IL-W procedure for material derivatives. The other main difference is that we need to construct the value of one more ghost point in each direction, which takes newly emerging points into account if necessary. Our method is shown to have the designed third order accuracy for problems with smooth solutions in both D and D, therefore more accurate than the finite volume schemes developed in [, 7, 9, 5]. In particular, a D accuracy test involving a complex moving geometry is proposed. Our method also performs well for the D moving piston problem and the cylinder lift-off problem.

24 0. Y X (a) h = / Y X (b) h = /80 Figure 4.: Pressure contours of Example 5, 5 contours from to 8. t =

25 0. Y X (a) h = / Y X (b) h = /80 Figure 4.4: Pressure contours of Example 5, 5 contours from to 8. t =

26 We only consider interactions between compressible inviscid flows and moving rigid bodies. In some applications, such as biological fluid dynamics, geometrically complicated structures are often deformable. Moreover, the fluid is usually considered to be incompressible or viscous. The effectiveness and robustness of our method for these types of problems are subject to future research. References [] M. Arienti, P. Hung, E. Morano and J.E. Shepherd, A level set approach to Eulerian- Lagrangian coupling, Journal of Computational Physics, 85 (00), -5. [] M.H. Carpenter, D. Gottlieb, S. Abarbanel and W.-S. Don, The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error, SIAM Journal on Scientific Computing, 6 (995), 4-5. [] J. Cheng and C.-W. Shu, A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, Journal of Computational Physics, 7 (007), [4] P. De Palma, M.D. de Tullio, G. Pascazio and M. Napolitano, An immersed-boundary method for compressible viscous flows, Computers & Fluids, 5 (006), [5] M.D. de Tullio, P. De Palma, G. Iaccarino, G. Pascazio and M. Napolitano, An immersed boundary method for compressible flows using local grid refinement, Journal of Computational Physics, 5 (007), [6] J. Falcovitz, G. Alfandary and G. Hanoch, A two-dimensional conservation laws scheme for compressible flows with moving boundaries, Journal of Computational Physics, 8 (997), 8-0. [7] H. Forrer and M. Berger, Flow simulations on Cartesian grids involving complex moving geometries, International Series of Numerical Mathematics, 9, Birkhäuser Verlag Basel/Switzerland, 999,

27 [8] R. Ghias, R. Mittal and H. Dong, A sharp interface immersed boundary method for compressible viscous flows, Journal of Computational Physics, 5 (007), [9] X.Y. Hu, B.C. Khoo, N.A. Adams and F.L. Huang, A conservative interface method for compressible flows, Journal of Computational Physics, 9 (006), [0] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 6 (996), 0-8. [] R. Mittal and G. Iaccarino, Immersed boundary methods, Annual Review of Fluid Mechanics, 7 (005), 9-6. [] S.M. Murman, M.J. Aftosmis and M.J. Berger, Implicit approaches for moving boundaries in a -D Cartesian method, AIAA Paper No. 00-9, 00. [] C.S. Peskin, Flow patterns around the heart valves, Journal of Computational Physics, 0 (97), 5-7. [4] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shockcapturing schemes, Journal of Computational Physics, 77 (988), [5] K.-M. Shyue, A moving-boundary tracking algorithm for inviscid compressible flow, Hyperbolic Problems: Theory, Numerics, Applications, Springer Berlin Heidelberg, 008, [6] S. Tan and C.-W. Shu, Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws, Journal of Computational Physics, 9 (00),