Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations Jianfang Lu

Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations

Jianfang Lu¹, Jinwei Fang², Sirui Tan³, Chi-Wang Shu⁴ and Mengping Zhang⁵ Abstract

We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficul- ties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure [28], in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, have been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this ex- tension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combi- nation of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive nu- merical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

Key Words: convection-diffusion equation, high order finite difference methods, nu- merical boundary condition, inverse Lax-Wendroff method, compressible Navier-Stokes equations

1School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: kobey@mail.ustc.edu.cn

2Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China.

E-mail: gerrard.fang@gmail.com

3Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:

sirui tan@alumni.brown.edu

4Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:

shu@dam.brown.edu. Research supported by AFOSR grant F49550-12-1-0399 and NSF grant DMS- 1418750.

5School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: mpzhang@ustc.edu.cn. Research supported by NSFC grant 11471305.

1 Introduction

The inverse Lax-Wendroff (ILW) method was introduced by Tan and Shu [28] as a technique of numerical boundary condition for solving hyperbolic equations on arbitrary geometry using high order finite difference methods. A stable and high order accurate numerical boundary treatment requires significant efforts for two reasons. First, the wide stencils of high order finite difference operators need special treatments for a few ghost points near the boundary. Second, the physical domain boundary in arbitrary geometry may not coincide with grid points and may intersect with the Cartesian mesh in an arbitrary fashion. There are several approaches to handle such boundary conditions. One commonly used method is to generate a boundary fitted mesh which allows us to impose boundary conditions directly in the algorithm. As a result, the governing equations are generally transformed into a new differential form in a curvilinear coordinate system (see e.g. [13]). If the domain is simple enough, a smooth mapping can be used for the transformation of the entire domain. But if the domain is complex, composite overlapping meshes are generated to fit the physical boundaries, while these meshes are connected via interpolation (see e.g. [4, 10, 9, 25]). The disadvantage of this method is the complexity of generating the body-conforming grids. Especially when the boundary varies with time, the generation of a new mesh at each time level is required, which increases the computational cost tremendously. Another approach is to use Cartesian grids which do not conform to the physical boundary. The embedded boundary method is developed to solve the wave equation with Dirichlet or Neumann boundary conditions by using finite difference methods on Cartesian grids in [15, 16, 14, 19]. In [24] the authors applied this method to hyperbolic conservation laws and obtained a second order accurate scheme. Baeza et al. [2] extended the approach from second order to fifth order using Lagrange extrapolation with a filter for the detection of discontinuities.

A third approach is the so called immersed boundary method, which makes a mod- ification of the original partial differential equations by introducing a forcing function

at the physical boundary, and uses it to reproduce the effects of boundary conditions, see, e.g. [22, 18]. The inverse Lax-Wendroff (ILW) method is similar to the immersed boundary method, but without introducing any forcing function to alter the original equations. In [28, 29, 30, 31], the authors developed this high order accurate boundary treatment for hyperbolic conservation laws, based on the ILW procedure for the inflow boundaries and high order extrapolation for the outflow boundaries. This boundary treatment allows us to compute hyperbolic conservation laws defined on an arbitrarily shaped domain covered by a Cartesian mesh with high order accuracy. The main idea of the ILW procedure is repeatedly using the partial differential equation to convert the normal derivatives into the time derivatives and tangential derivatives at the physical boundary. With the given inflow boundary condition and these normal derivatives, we can use Taylor expansions to assign values to ghost points near the physical boundary.

A simplified ILW procedure which uses the ILW process only for the first few normal derivatives and then the less expensive high order extrapolation for the remaining ones is developed in [31]. Stability analysis for both the ILW and the simplified ILW proce- dures is given in [32, 17], proving that the method remains stable under the same CFL condition as that for problems without boundaries, regardless of the location of the first grid point relative to the location of the physical boundary. This indicates that the “cut cell” problem [1], which refers to the instability or the requirement of the extremely small CFL condition when the first grid point does not coincide with but is very close to the physical boundary, is effectively removed by the ILW or the simplified ILW procedure.

For earlier related work, see [6, 7, 11].

So far, the ILW method has been mostly used on hyperbolic equations including con- servation laws [28, 29, 31, 17] and Boltzmann type models [5], which involve only first or- der spatial derivatives in the equations. In this paper, we attempt to extend this method- ology to convection-diffusion equations. It turns out that this extension is non-trivial, as totally different boundary treatments are needed for the diffusion-dominated and the

convection-dominated regimes. A careful combination of these two boundary treatments is designed in this paper to obtain a stable and accurate boundary condition for high or- der finite difference schemes when applied to convection-diffusion equations, regardless of the regimes. We provide extensive numerical tests for one and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

We comment on the additional issues encountered in solving the compressible Navier- Stokes equations. Unlike the model convection-diffusion equations that we consider first, the compressible Navier-Stokes equations form a nonlinear incompletely parabolic system. There are many papers discussing boundary conditions of incompletely parabolic systems based on the energy method, for example [8, 21, 20]. Our approach for the diffusion-dominated regime of the compressible Navier-Stokes equations is based on the theory in [26, 8], which reduces the incompletely parabolic system into a parabolic system and a strictly hyperbolic system.

This paper is organized as follows. In Section 2, we first illustrate our idea by devel- oping our inverse Lax-Wendroff method for one-dimensional scalar convection-diffusion equations. We then generalize it to one-dimensional systems and two-dimensional prob- lems. For the convection-dominated cases with a shock-like sharp transition region near the outflow boundary, we adopt a weighted essentially non-oscillatory (WENO) type extrapolation, see e.g. [28, 31, 12], to maintain nonlinear stability. In Section 3, we show numerical results to demonstrate the effectiveness and robustness of our approach.

Concluding remarks are given in section 4.

2 Scheme formulation

We consider a convection-diffusion equation, abstractly written as

U_t=L(U), (2.1)

whereLis a spatial operator involving first and second order spatial derivatives. After the spatial operator is discretized by a high order finite difference operator, the semi-discrete scheme is written abstractly as the following ordinary differential equation (ODE) system

U_t=Lh(U). (2.2)

Our main interest is in the convection-dominated regime. Thus we use the following third order total variation diminishing (TVD) Runge-Kutta method [23] to discretize (2.2)

U⁽¹⁾ =Uⁿ+ ∆tLh(Uⁿ), U⁽²⁾ = 3

4Uⁿ+1

4U⁽¹⁾+1

4∆tLh(U⁽¹⁾), Uⁿ⁺¹ = 1

3Uⁿ+2

3U⁽²⁾+2

3∆tLh(U⁽²⁾).

In [3], the authors pointed out that the boundary condition should be imposed as follows in order to maintain the third order accuracy

Uⁿ =g(tn),

U⁽¹⁾ =g(tn) + ∆tg^′(tn), U⁽²⁾ =g(t_n) + ∆t

2 g^′(t_n) + ∆t²

4 g^′′(t_n),

where g(t) is the boundary condition for the governing equations. We follow this ap- proach in our numerical boundary treatments. For simplicity, in the following we simply use g(t) to denote our boundary condition, without distinguishing its different meaning at different Runge-Kutta stages.

2.1 Inverse Lax-Wendroff method for hyperbolic equations

Let us briefly review the ILW method for hyperbolic equations [28], to explain the basic idea and to set notations. For simplicity, we consider the following one-dimensional linear equation







ut+ux = 0, −1< x <1, t >0 u(x,0) =u0(x), −1≤x≤1 u(−1, t) =g(t), t >0.

(2.3)

For two constants δ1, δ2 ∈ [0,1), we use a uniform mesh with the mesh size ∆x =

2

N+δ1+δ2 and distribute the mesh points as

−1 +δ1∆x=x0 < x1 <· · ·< xN = 1−δ2∆x. (2.4) Notice that we have deliberately allowed the physical boundary x = ±1 not coinciding with grid points. We take {x0, x1,· · ·, xN} as our interior points.

In this paper, for simplicity we use the third order upwind-biased conservative finite difference operator (or the corresponding third order WENO operator [12]) to approx- imate the first order spatial derivative, unless otherwise stated. The methodology cer- tainly also works for other high order finite difference operators. Forj = 0,1,· · · , N, we have the semi-discrete scheme

(u_j)_t+ 1

∆x(ˆu_j+1

2 −uˆ_j−1

2) = 0, (2.5)

with the numerical flux ˆu_j+¹

2 defined as ˆ

u_j+¹

2 =−1

6uj−1+5

6uj+ 1

3uj+1. (2.6)

Hereuj is the numerical approximation to the exact solutionu at the grid point (xj, t).

The scheme (2.5) requires a four point stencil so up to two ghost points are needed near the boundaries. We concentrate on describing how to define the values at the ghost points {x−2, x−1, xN+1}.

Our goal is to obtain spatial derivatives of each order at the physical boundary, then use Taylor expansion to get the values at the ghost points. The Taylor expansion ofs-th order at the left and right boundary is respectively defined as

uj =

s

X

k=0

(xj + 1)^k

k! ∂_x^(k)u, j =−2,−1, (2.7a) u_j =

s

X

k=0

(xj −1)^k

k! ∂_x^(k)u, j =N + 1. (2.7b)

Here ∂x^(k)u is the numerical approximation of the k-th spatial derivative of u at the physical boundaryx=−1 in (2.7a) or atx= 1 in (2.7b). For the left boundary, we can

use the equation and the boundary condition to get

∂_x⁽⁰⁾u=u(−1, t) = g(t),

∂_x⁽¹⁾u=ux(−1, t) =−ut(−1, t) =−g^′(t),

∂_x⁽²⁾u=uxx(−1, t) =−utx(−1, t) =utt(−1, t) =g^′′(t),

· · ·

Thus we can obtain{∂^(k)x u, k = 0,1,· · · }completely from the given boundary condition and the partial differential equation (PDE) by converting the spatial derivative into the time derivatives. This is very similar to the traditional Lax-Wendroff scheme, in which the time derivatives are rewritten in terms of spatial derivatives through repeatedly using the PDE. Here the roles of time and space are reversed, hence the method was referred to as the inverse Lax-Wendroff procedure. In the following, we use the superscript

“ilw” to denote derivatives obtained through the ILW procedure, e.g. u^ilw_x means that ux is derived from the ILW procedure. Even though this ILW procedure works for general inflow case for nonlinear problems, multi-dimensions and systems, the algebra could be quite complicated, especially for higher order spatial derivatives. Moreover, this procedure certainly does not apply to the outflow boundary, where no physical boundary condition is provided.

Another way to obtain approximation to these spatial derivatives at the boundary is through the traditional Lagrangian or WENO extrapolation [28, 31] from interior points with suitable order of accuracy. We use the superscript “ext” to denote derivatives obtained through the extrapolation procedure. Extrapolation is needed not only at the outflow boundary, but also at the inflow boundary for as many higher order spatial derivative terms as stability allows. In [31], a simplified ILW procedure is introduced to use the ILW procedure only to obtain the solution and first order spatial derivative at the inflow boundary, and use extrapolation to obtain all other derivatives. This gives a stable fifth order WENO algorithm when the physical point is aligned with the grid.

In [32, 17], systematic analyses are performed for central compact schemes and upwind- biased finite difference schemes respectively for the simpled ILW procedure, to determine the minimum number of spatial derivatives needed from the ILW procedure in order to guarantee stability, regardless of the distance from the physical boundary to the closest grid point.

2.2 One-dimensional scalar convection-diffusion equations

We consider the following one-dimensional scalar convection-diffusion equation







ut+f(u)x =εuxx, −1< x < 1, t > 0, u(−1, t) =g1(t), u(1, t) = g2(t), t >0 u(x,0) =u0(x), −1≤x≤1

(2.8)

where ε > 0 is a constant. Without loss of generality, we assume f^′(g1(t)) > 0 and f^′(g2(t))≥0.

We still use the uniform mesh defined in (2.4). The first order spatial derivative is approximated by the third order upwind-biased finite difference operator (or its WENO extension), and the second order spatial derivative is approximated by the central fourth order discretization. Forj = 0,1,· · ·, N, we have the semi-discrete scheme

(uj)t+ 1

∆x( ˆfj+1/2−fˆj−1/2) = ε

12∆x²(−uj+2+ 16uj+1−30uj+ 16uj−1−uj−2) (2.9) where ˆfj+1/2 is the upwind-biased numerical flux. For instance, if f^′(u)≥0, we have

fˆ_j+1

2 =−1

6f(u_j−1) + 5

6f(u_j) + 1

3f(u_j+1)

Our goal is to define the values at the ghost points {x−2, x−1, xN+1, xN+2}.

We first consider the left boundary x=−1. We can always use the given boundary condition to get u = g1(t) in the Taylor expansion, hence the problem is to define the first and higher order derivatives in the Taylor expansion. As explained before, we have two choices. One is to extrapolate from the interior points; the other is to use the ILW method to get the derivatives from the equation and the given boundary condition.

We start with two extreme cases: (1) f^′(g1(t)) = 0, i.e., pure diffusion, and (2) ε= 0, i.e., pure convection. For the pure diffusion case with f^′(g1(t)) = 0, we can only obtain the second order (or higher even order) spatial derivative u^ilw_xx = ^g′1_ε^(t) using the ILW method. The first order (or higher odd order) spatial derivative has to come from extrapolation. For the purely convection case with ε= 0, we can obtain the first order (or any higher order) spatial derivative u^ilw_x = _f′^g_(g^1′(t)

1(t)) by the ILW method, as in the hyperbolic case. To save the cost, the second order spatial derivative is obtained using extrapolation according to the result in [31], as explained before.

In the general case when f^′(g1(t))6= 0 and ε 6= 0, we have two ways to perform the ILW procedure. We may obtain the second order spatial derivative by extrapolation, then obtain the first order spatial derivative through the ILW procedure

u^ilw_x = 1

f^′(g1(t))(−g^′₁(t) +εu^ext_xx),

which is what we would do for the purely convection case and thus is expected to work in the convection-dominated regime. Alternately, we may obtain the first order spatial derivative by extrapolation, then obtain the second order spatial derivative through the ILW procedure

u^ilw_xx = 1

ε(g₁^′(t) +f^′(g1(t))u^ext_x ),

which is what we would do for the purely diffusion case and therefore is expected to work in the diffusion-dominated regime.

In order to obtain a strategy which is applicable for all convection-diffusion equations, we form a convex combination of the two procedures described above

ux =αu^ilw_x + (1−α)u^ext_x , uxx= (1−α)u^ilw_xx +αu^ext_xx,

where 0 ≤ α ≤ 1 satisfies the following conditions: α = 0 for the pure diffusion case with f^′(g1(t)) = 0; and α = 1 for the pure convection case with ε = 0. After extensive numerical tests, we have settled on the choice

α= f^′(g1(t))²∆x²

f^′(g1(t))²∆x²+ 9ε². (2.10)

Extending the strategy to all the spatial derivatives, we have







∂x⁽⁰⁾u

∂x⁽¹⁾u

∂x⁽²⁾u





= f^′(g1(t))²∆x² f^′(g1(t))²∆x²+ 9ε²



 g1(t)

u^ilw_x u^ext_xx



+ 9ε²

f^′(g1(t))²∆x² + 9ε²



 g1(t)

u^ext_x u^ilw_xx



. (2.11) Using these values, we can obtain the values of the numerical solution at the ghost points near the left boundary by the Taylor expansion (2.7a). The same procedure can be used on the right boundary x= 1.

2.3 One-dimensional linear systems

We now extend the numerical boundary condition to one-dimensional linear systems.

We consider the following equations

U_t+AU_x =BU_xx, −1< x < 1, t >0 (2.12) with appropriate initial and boundary conditions. Here

U = (U₁, U₂,· · · , U_d)^T,

AandB are constantd×dmatrices. The eigenvalues ofAare real andAhas a complete set of eigenvectors. B is a positive-definite matrix. We still have the uniform mesh de- fined in (2.4). Our goal is to define the values at the ghost points{x−2, x−1, xN+1, xN+2}.

The main complication for the system case is that the convection term and the diffusion term in general cannot be simultaneously diagonalized. This makes a direct application of the strategy for the scalar convection-diffusion equation difficult. Without loss of generality, we only consider the left boundary.

2.3.1 Convection-dominated case

In the convection-dominated case, we first diagonalize the matrixAand obtain its eigen- values{λ1, λ2,· · · , λd}and the corresponding left eigenvectors{l1,l₂,· · · ,l_d}to form the matrix

L=







 l₁ l₂

· · · l_d









=









l_1,1 l_1,2 · · · l_1,d l2,1 l2,2 · · · l2,d

· · · · ld,1 ld,2 · · · ld,d







 .

Then we have LAL⁻¹ = Λ = diag(λ1, λ2,· · · , λd). With this characteristic decomposi- tion V =LU, the equations (2.12) becomes

V_t+ ΛV_x =LBU_xx =RHS^V. (2.13)

Since B is a positive-definite matrix, we should have d boundary conditions at the boundary for well-posedness. However, in the convection-dominated case, we impose characteristic boundary conditions for V, depending on the signs of the eigenvalues in Λ.

We look at the specific case of{λj ≤0, j = 1,2,· · · , kn}and{λj >0, j =kn+ 1, kn+ 2,· · ·, d}. The characteristic variables {V_j, j = 1,· · · , k_n}have outflow boundary condi- tion at x=−1, and{Vj, j =kn+ 1,· · ·, d} have inflow boundary conditions atx=−1.

We denote∂x^(k)U_cas the numerical approximation to ^∂_∂x^kUk on the boundary. The subscript

“c” indicates the convection-dominated case. For j = 1,· · · , kn, the outgoing charac- teristic variable Vj can be extrapolated from interior point values {(Vj)i, i = 0,1,2,3}

to the boundary to get {(Vj)^ext,(Vj)^ext_x ,(Vj)^ext_xx}. For j = kn + 1,· · · , d, the incoming characteristic variable is set by the given boundary condition as Vj = (Lg)j(t), where g(t) is the given boundary condition for U and (Lg)_j(t) is thej-th entry of the vector Lg. With the extrapolation values and boundary conditions we have

(

l_j·∂x⁽⁰⁾U_c = (Vj)^ext, j= 1,· · · , kn

l_j·∂x⁽⁰⁾U_c = (Lg)j(t), j =kn+ 1,· · · , d (2.14) Next we would like to find the first order derivatives ∂x⁽¹⁾U by the inverse Lax- Wendroff procedure, which is the essential part of the algorithm in [28, 31]. Since the boundary conditions for {V_j, j = k_n + 1,· · · , d} are prescribed, combining with the information of outgoing characteristic variable {Vj, j = 1,· · · , kn}, we have

(

l_j·∂x⁽¹⁾U_c = (Vj)^ext_x , j= 1,· · · , kn

(LA)j ·∂x⁽¹⁾U_c =−(Lg)^′_j(t) +RHS_j^V, j =kn+ 1,· · · , d (2.15) where (LA)m is the m-th row vector of the matrix LA and RHS_m^V is the m-th entry of the vectorRHS^V in (2.13). All second order derivativesU_xx inRHS^V are obtained by

extrapolation. After we obtain∂x⁽¹⁾U_c, the second order derivatives ∂x⁽²⁾U_c are obtained by extrapolation.

2.3.2 Diffusion-dominated case

We now turn to the diffusion-dominated case. Firstly we diagonalize the matrix B and obtain its eigenvalues{λ^′₁, λ^′₂,· · · , λ^′_d}and the corresponding left eigenvectors{l^′₁,l^′₂,· · · ,l^′_d} to form the matrix

L^′ =







 l^′₁ l^′₂

· · · l^′_d









=









l^′_1,1 l^′_1,2 · · · l_1,d^′ l^′_2,1 l^′_2,2 · · · l_2,d^′

· · · · l^′_d,1 l^′_d,2 · · · l_d,d^′









Then we have L^′BL^′−1 = Λ^′ = diag(λ^′₁, λ^′₂,· · · , λ^′_d). Since the matrix B is positive- definite, the eigenvalues are positive. With this characteristic decompositionW =L^′U, the equations (2.12) becomes

W_t+CWx = Λ^′W_xx, (2.16)

where C=L^′AL^′−1. Our algorithm to get the derivatives of each order is as follows

∂⁽⁰⁾_x U_d=g(t), (2.17)

∂⁽¹⁾_x U_d=U_x^ext, (2.18)

∂⁽²⁾_x U_d=B⁻¹(−g^′(t) +AU_x^ext), (2.19) where ∂x^(k)U_d is the numerical approximation to ^∂_∂x^kUk in the diffusion-dominated case.

2.3.3 General convection-diffusion case

Now we combine the convection- and diffusion-dominated cases together. In order to obtain the convex combination coefficient, we would like to use the matricesC and Λ^′ in (2.16). First we define the coefficients for both the convection- and diffusion-dominated cases as follows

aj = ∆x²

d

X

i=1

C_j,i² , εj = 9λ^′_j², αj = aj

aj +εj

, j = 1,2,· · ·, d, (2.20)

where Cj,i is the (j, i) entry of matrix C in (2.16). We then consider equation (2.16), and take a transformation of∂x^(k)U_c and ∂x^(k)U_d as

∂_x^(k)W_c =L^′∂_x^(k)U_c, ∂_x^(k)W_d=L^′∂_x^(k)U_d, k = 0,1,2, (2.21) and make a convex combination of them

∂_x^(k)Wj =αj∂_x^(k)(Wc)j + (1−αj)∂_x^(k)(Wd)j, k= 0,1,2. (2.22) After we get {∂x^(k)W, k= 0,1,2}, we transform back and obtain

∂_x^(k)U =L^′−1∂_x^(k)W, k= 0,1,2. (2.23) With these derivatives, we can use Taylor expansion to get values of the numerical solution at the ghost points.

We summarize our algorithm of the boundary treatment as follows, under the as- sumption that {Uj, j = 0,1,· · · , N} have been updated at time level tn. Here we just take the left boundary as an example to illustrate our algorithm.

Algorithm 1:

1. For the convection-dominated case:

• Compute the eigenvalues of A to decide the inflow and outflow boundary conditions. Use the characteristic decomposition ofAand obtain the equation (2.13).

• For each outgoing characteristic variable Vj, we use extrapolation to obtain {(Vj)^ext,(Vj)^ext_x ,(Vj)^ext_xx}. With the prescribed boundary conditions for the in- coming characteristic variables and extrapolated values (V_j)^ext, we can obtain

∂x⁽⁰⁾U_c by solving the linear system (2.14).

• With the inverse Lax-Wendroff procedure for the incoming characteristic vari- ables and extrapolated values (Vj)^ext_x , we can obtain ∂x⁽¹⁾U_c by solving the linear system (2.15).

• The second order derivative is obtained by extrapolation, i.e. ∂x⁽²⁾U_c =U_xx^ext. 2. For the diffusion-dominated case:

• Diagonalize the matrix B and obtain the equation (2.16).

• Impose boundary conditions for ∂x⁽⁰⁾U_d on the boundary, as in (2.17).

• The first order derivative ∂x⁽¹⁾U_d is obtained by extrapolation, as in (2.18).

• Use the inverse Lax-Wendroff method to obtain the second order derivative

∂x⁽²⁾U_d, as in (2.19).

3. For the general convection-diffusion case:

• Choose the coefficient {αj, j = 1,· · · , d}defined in (2.20).

• Convert ∂x^(k)U_c and ∂x^(k)U_d into ∂x^(k)W_c and ∂x^(k)W_d, k = 0,1,2 respectively, as in (2.21).

• Make a convex combination of the derivatives ∂x^(k)W_c and ∂x^(k)W_d to get the derivatives ∂x^(k)W, k= 0,1,2, as in (2.22).

• Transform from ∂x^(k)W back to∂x^(k)U, k = 0,1,2, as in (2.23).

4. Assign the values of the ghost points by the Taylor expansion (2.7).

2.4 One-dimensional Navier-Stokes equations

We consider the one-dimensional compressible Navier-Stokes equations in the non-dimensional form [33]







ρ_t+ (ρu)_x= 0

(ρu)t+ (ρu² +p)x = _Re^{1 4}₃u

xx

Et+ (u(E+p))x = _Re¹ h

2

3(u²)xx+ _(γ−¹_{1)P r}(c²)xx

i (2.24)

with appropriate initial and boundary conditions. Hereρis the density,uis the velocity, E is the total energy,p is the pressure, the equation of state is

E = p

γ−1+ 1 2ρu²,

with γ = 1.4 for air. The speed of sound is given as c=

rγp ρ .

Reis the Reynolds number, and P r= 0.7 is the Prandtl number.

Compared with one-dimensional linear system, there are two more features of the Navier-Stokes equations. The first one is nonlinearity. The second one is that there is no diffusion term in the density equation. In order to motivate our choice of the numerical boundary condition, we rewrite the system into the following form for smooth solutions U_t+A(U)Ux =B(U)Uxx+N LT, (2.25) where

U = (U₁, U₂, U₃)^T = (ρ, ρu, E)^T,

A(U) =











0 1 0

1

2(γ−3)u² (3−γ)u γ−1 1

2(γ −1)u³−uH H −(γ−1)u² γu









 ,

B(U) = 1 Re













0 0 0

−4u 3ρ

4

3ρ 0

γ P r − 4

3 u²

ρ − γ P r

E ρ²

4 3− γ

P r u

ρ γ P rρ











 ,

and the additional nonlinear term is given by

N LT = 1 Re











0 8

3ρ³

−ρρx(ρu)x+ρ²_xρu 1

ρ² 4

3 − γ P r

(3uρ_x−(ρu)_x)(uρ_x−(ρu)_x) + 2γ P rρ³

−ρρ_xE_x+ρ²_xE









 .

Here, H is enthalpy defined by

H = E+p ρ . 2.4.1 Convection-dominated case

We first consider the convection-dominated case, i.e., when the Reynolds number Re is large. We treat this case along the lines described in the previous subsection and in

the same way as in [31] for the one-dimensional Euler equations. Namely, we apply the inverse Lax-Wendroff procedure for the first order derivative terms, and use extrapolation to obtain all the second order derivatives terms, including those in the nonlinear term.

Considering the left boundary x = −1, we first determine the boundary value U_b through the given boundary condition and extrapolation for the variables not given by the boundary condition. We then use this boundary value to diagonalize the convection term A(Ub) and obtain its three eigenvalues λ1 = u−c, λ2 = u, λ3 = u +c and the corresponding left eigenvectors {l1,l₂,l₃} which form the matrix

L=



 l₁ l₂ l₃



=





l1,1 l1,2 l1,3

l2,1 l2,2 l2,3

l3,1 l3,2 l3,3



.

Thus we have LAL⁻¹ = Λ = diag(λ1, λ2, λ3) on the left boundary x = −1. With this characteristic decomposition V = LU, the equations (2.25) on the left boundary becomes

V_t+ ΛV_x =L(B(U)Uxx+N LT) =RHS^V. (2.26) The number of given boundary conditions depends on the number of positive eigenvalues in Λ. This is the same as in Euler equations, although the number of given boundary conditions for the Navier-Stokes equations is always equal to or more than that for the Euler equations. In the case that the number of given boundary conditions for the Navier- Stokes equations is more than that for the Euler equations, we take the given boundary condition for the convection-dominated case sequentially by the following ordered list:

E, ρuand ρ, until the required number is reached.

We now look at the specific case of λ1 ≤ 0, λ2 > 0, λ3 > 0. The characteristic variable V1 has an outflow boundary condition at x = −1, and V2, V3 have inflow boundary conditions at x = −1. For simplicity, we denote ∂x^(k)U_c as the numerical approximation to ^∂_∂x^kUk at the boundary. By the ordered preference list above, we should impose ∂x⁽⁰⁾(Uc)2 =g2(t) and ∂x⁽⁰⁾(Uc)3 =g3(t) using the given boundary conditions. For the outgoing characteristic variable V1, we extrapolate from {(V1)i, i = 0,1,2,3} to the

boundary to get {(V1)^ext,(V1)^ext_x ,(V1)^ext_xx}. Finally, ∂x⁽⁰⁾(Uc)1 can be obtained from the following equations







∂x⁽⁰⁾(Uc)2 =g2(t),

∂x⁽⁰⁾(Uc)3 =g3(t), l₁·∂x⁽⁰⁾U_c = (V1)^ext.

(2.27)

Next we would like to find the first order derivatives ∂x⁽¹⁾U_c by the inverse Lax-Wendroff procedure. Since the boundary conditions forU2 and U3 are prescribed, combining with the information of outgoing characteristic variableV1, we have







l₁·∂x⁽¹⁾U_c = (V₁)^ext_x ,

A2·∂x⁽¹⁾U_c =−g₂^′(t) +RHS₂^V, A3·∂x⁽¹⁾U_c =−g₃^′(t) +RHS₃^V,

(2.28)

where Am is the m-th row vector of the matrix A and RHS_m^V is the m-th entry of the vector RHS^V in (2.26), m = 1,2,3. All derivatives in RHS^V are obtained by extrapolation. After we obtain∂x⁽¹⁾U_c, the second order derivatives ∂x⁽²⁾U_c are obtained by extrapolation.

2.4.2 Diffusion-dominated case

We now turn to the diffusion-dominated case, i.e., when Re is small. Special attention must be paid to the fact that this is an incompletely parabolic system. In [26, 8], the authors decoupled this incompletely parabolic system into a strictly hyperbolic system and a parabolic system, and then deduced sufficient conditions for the well-posedness of the incompletely parabolic system. We take this decoupling as a guidance for the design of our algorithm in the diffusion-dominated case. We diagonalize the matrix in front of the second order derivative terms on the boundary to obtain

W_t+CWx = Λ^′W_xx+L^′N LT, (2.29) where

W =L^′U, L^′BL^′−1 = Λ^′ = 1 Reρdiag

0,4

3, γ P r

,

and

C =











u −c

γ 0

−c u −c

0 −γ−1

γ c u











, L^′ =



 l^′₁ l^′₂ l^′₃



=





l^′_1,1 l^′_1,2 l_1,3^′ l^′_2,1 l^′_2,2 l_2,3^′ l^′_3,1 l^′_3,2 l_3,3^′



,

where {l^′₁,l^′₂,l^′₃} forms a complete set of left eigenvectors of the matrix B(U) on the boundary. We consider the linear part of (2.29), W_t +CW_x = Λ^′W_xx, which is an incomplete parabolic equations. In [8], they separated this linear system into

(W1)t+C1,1(W1)x = 0, W2

W₃

t

+

C2,2 C2,3

C_3,2 C_3,3

W2

W₃

x

= 1

Reρ



 4

3 0

0 γ

P r



 W2

W₃

xx

.

This is a combination of a scalar hyperbolic equation and a parabolic system. With suitable initial and boundary conditions, they proved that this system is well-posed.

This motivates us to treat the original system as a combination of a hyperbolic equation and a parabolic system. Let us consider two cases: u > 0 and u ≤ 0 on the left boundary x =−1. If u > 0, then we have three given boundary conditions g(t) on the left boundary. Our algorithm to get the derivatives of each order is as follows

∂_x⁽⁰⁾U_d =g(t), (2.30)







A1·∂x⁽¹⁾U_d=−g₁^′(t) +B1·U_xx^ext+NLT1, l^′₂·∂x⁽¹⁾U_d = (W2)^ext_x ,

l^′₃·∂x⁽¹⁾U_d = (W3)^ext_x ,

(2.31)







l₁^′ ·∂x⁽²⁾U_d= (W1)^ext_xx,

B₂ ·∂x⁽²⁾U_d=g^′₂(t) +A₂·U_x^ext−NLT₂, B3 ·∂x⁽²⁾U_d=g^′₃(t) +A3·U_x^ext−NLT3,

(2.32)

where B_m is the m-th row vector of the matrix B, m = 1,2,3, and the derivatives in N LT in (2.31) and (2.32) are obtained by extrapolation.

For u ≤ 0, we only have two given boundary conditions. Since the second order derivatives appear in the momentum and energy equations in (2.24), we takeρu, E as the

variables with given boundary conditions, and ρ is obtained by extrapolation. Namely, the equations (2.30) and (2.31) are replaced by (2.33) and (2.34) to obtain ∂x⁽⁰⁾U_d and

∂x⁽¹⁾U_d, while (2.32) stays the same.







∂⁽⁰⁾x (Ud)1 = (U1)^ext,

∂⁽⁰⁾x (Ud)2 =g2(t),

∂⁽⁰⁾x (Ud)3 =g3(t),

(2.33)

∂_x⁽¹⁾U_d =U_x^ext. (2.34)

2.4.3 General convection-diffusion case

Now we combine the two cases together. In order to obtain the convex combination coefficient, we would like to use the matrices C and Λ^′ in (2.29). First we define the coefficients for both the convection-dominated case and the diffusion-dominated case as follows

a1 = (C_1,1² +C_1,2² +C_1,3² )∆x², ε1 = 9

2(Λ^′_2,2² + Λ^′_3,3² ), α1 = a1

a1+ε1

, a2 = (C_2,1² +C_2,2² +C_2,3² )∆x², ε2 = 9Λ^′_2,2² , α2 = a2

a2+ε2

, (2.35)

a₃ = (C_3,1² +C_3,2² +C_3,3² )∆x², ε₃ = 9Λ^′_3,3² , α₃ = a₃ a3+ε3

.

We then consider equation (2.29), and take a transformation of ∂x^(k)U_c and ∂x^(k)U_d, k = 0,1,2 as

∂_x^(k)W_c =L^′∂_x^(k)U_c, ∂_x^(k)W_d =L^′∂_x^(k)U_d, k = 0,1,2 (2.36) and make a convex combination of them

∂_x^(k)Wi =αi∂^(k)_x (Wc)i+ (1−αi)∂_x^(k)(Wd)i, i= 1,2,3, k= 0,1,2. (2.37) Notice that we have taken ε1 as the average of ε2 and ε3 because Λ^′_1,1 is always 0. If we took ε1 = 9Λ^′_1,1² , we would not be able to recover the algorithm for the diffusion- dominated case because ∂x^(k)W1 would always equal to ∂x^(k)(Wc)1. After we obtain

∂x^(k)W, k = 0,1,2, we transform back and have

∂_x^(k)U =L^′−1∂_x^(k)W, k= 0,1,2. (2.38)

With these derivatives, we can use Taylor expansion (2.7a) to impose values of the numerical solution at the ghost points.

Let us summarize our algorithm of the boundary treatment for the Navier-Stokes equations as follows, assuming that {U_j, j = 0,1,· · ·, N} have been updated at time levelt_n. We only consider the left boundary here, since there is no difference in applying this method to the right boundary.

Algorithm 2:

1. For the convection-dominated case:

• Compute the eigenvalues of A in (2.25) to decide the inflow and outflow boundary conditions. Use the characteristic decomposition of A and obtain the equation (2.26).

• For each outgoing characteristic variable Vj, we use extrapolation to obtain {(Vj)^ext,(Vj)^ext_x ,(Vj)^ext_xx}. With the prescribed boundary conditions for the in- coming characteristic variables and extrapolated values (Vj)^ext, we can obtain

∂x⁽⁰⁾U_c by solving a linear system, such as (2.27).

• With the inverse Lax-Wendroff procedure for the incoming characteristic vari- ables and extrapolated values (Vj)^ext_x , we can obtain∂⁽¹⁾x U_c by solving a linear system, such as (2.28).

• The second order derivatives are obtained by extrapolation, i.e. ∂x⁽²⁾U_c = U_xx^ext.

2. In the diffusion-dominated case, we have two different cases depending on the sign of u. Here we consider u >0.

• We have three boundary conditions for ∂x⁽⁰⁾U_d, as in (2.30).

• The first order derivative ∂⁽¹⁾x U_d are obtained by applying the inverse Lax- Wendroff method to the mass equation, together with the extrapolation equa- tions for ∂x⁽¹⁾(Wd)2 and ∂x⁽¹⁾(Wd)3, as in (2.31).

• The second order derivative∂x⁽²⁾W_d are obtained by applying the inverse Lax- Wendroff method to momentum equation and energy equation, together with the extrapolation equation for ∂x⁽²⁾(Wd)1, as in (2.32).

3. For the general case, to combine ∂x^(k)U_c and ∂^(k)x U_d, k= 0,1,2:

• Choose the coefficient {α_j, j = 1,2,3} defined in (2.35).

• Convert ∂x^(k)U_c and ∂x^(k)U_d into ∂x^(k)W_c and ∂x^(k)W_d, k = 0,1,2, respectively, as in (2.36).

• Make a convex combination of the derivatives ∂^(k)x W_c and ∂^(k)x W_d to obtain the derivatives ∂x^(k)W, k = 0,1,2, as in (2.37).

• Transform from ∂x^(k)W back to∂x^(k)U, k = 0,1,2, as in (2.38).

4. Assign the values of the ghost points by the Taylor expansion (2.7a).

Remark: If u ≤ 0, we replace (2.30) and (2.31) by (2.33) and (2.34) to obtain ∂x⁽⁰⁾U_d and ∂x⁽¹⁾U_d in Step 2, while other steps remain the same.

2.5 Two-dimensional Navier-Stokes equations

We consider the two-dimensional non-dimensionalized compressible Navier-Stokes equa- tions (see e.g. [27, 33])

U_t+F(U)x+G(U)y = 1

Re(S₁(U)x+S₂(U)y), (x, y)∈Ω, t >0, (2.39) with suitable initial and boundary conditions, where

U = (U1, U2, U3, U4)^T = (ρ, ρu, ρv, E)^T,

F(U) = (ρu, ρu²+p, ρuv, u(E+p))^T, G(U) = (ρv, ρuv, ρv²+p, v(E+p))^T, S₁(U) = (0, τ11, τ21, σ1)^T,

S₂(U) = (0, τ12, τ22, σ2)^T,

with the components of the viscous stress tensor given by τ₁₁= 4

3u_x− 2

3v_y, τ₁₂ =τ₂₁ =u_y +v_x, τ₂₂= 4 3v_y− 2

3u_x. and

σ1 =uτ11+vτ12+ (c²)x

(γ−1)P r, σ2 =uτ21+vτ22+ (c²)y

(γ−1)P r.

As in the one-dimensional case,Reis the Reynolds number,γ = 1.4 for air, andP r= 0.7 is the Prandtl number. We also have

E = p

γ−1+ 1

2ρ(u²+v²), c= rγp

ρ , H = E+p ρ .

To define the value at a ghost point P = (xi, yj), we find a point P0 = (x0, y0) on the physical boundary ∂Ω such that the outward normal vector n at P0 goes through the ghost point P. For the ease of the inverse Lax-Wendroff method, we set up a local coordinate system at P₀ by

xˆ ˆ y

=T x

y

, T =

cosθ sinθ

−sinθ cosθ

, (2.40)

where θ is the angle between n and the x-axis and T is the rotation matrix. After this transformation we have ˆxsharing the same direction with the normal n and ˆy becomes the tangential direction of ∂Ω at P0. If we take

uˆ ˆ v

=T u

v

, then we get a new system

Uˆ_t+F( ˆU)xˆ+G( ˆU)yˆ= 1

Re(S₁( ˆU)xˆ+S₂( ˆU)yˆ) (2.41)

with the new variables

Uˆ = ( ˆU1,Uˆ2,Uˆ3,Uˆ4)^T = (ρ, ρˆu, ρˆv, E)^T. (2.42) We concentrate on this system, which is the same as the original Navier-Stokes system (2.39) because of rotational invariance. Our goal is to obtain{∂_x^(k)_ˆ Uˆ,k = 0,1,2}, which is a numerical approximation to ^∂_∂ˆ^k_x^Uˆk at P0. First we rewrite the system (2.41) as

Uˆ_t+A( ˆU) ˆU_xˆ+B( ˆU) ˆU_yˆ =C( ˆU) ˆU_ˆxˆx+D( ˆU) ˆU_yˆˆy+E( ˆU) ˆU_xˆˆy +N LT (2.43) where

A( ˆU) =









0 1 0 0

(γ−1)q−uˆ² (3−γ)ˆu (1−γ)ˆv γ−1

−ˆuˆv vˆ uˆ 0

−ˆuH+ (γ−1)ˆuq H+ (1−γ)ˆu² (1−γ)ˆuˆv γuˆ









with

q = 1

2(ˆu²+ ˆv²) We also have

B( ˆU) =









0 0 1 0

−ˆuˆv vˆ uˆ 0

(γ−1)q−vˆ² (1−γ)ˆu (3−γ)ˆv γ−1

−ˆvH+ (γ−1)ˆvq (1−γ)ˆuˆv H+ (1−γ)ˆv² γˆv







 ,

C( ˆU) = 1 Reρ









0 0 0 0

−⁴₃uˆ ⁴₃ 0 0

−ˆv 0 1 0

−⁴₃uˆ²−vˆ²+ _{P r}^γ q− _{(γ−1)P r}^c2 (⁴₃ − _{P r}^γ )ˆu (1− _{P r}^γ )ˆv _{P r}^γ







 ,

D( ˆU) = 1 Reρ









0 0 0 0

−ˆu 1 0 0

−⁴₃vˆ 0 ⁴₃ 0

−ˆu²− ⁴₃ˆv²+ _{P r}^γ q−_{(γ−1)P r}^c2 (1− _{P r}^γ )ˆu (⁴₃ −_{P r}^γ )ˆv _{P r}^γ







 ,

and

E( ˆU) = 1 3Reρ









0 0 0 0

−ˆv 0 1 0

−ˆu 1 0 0

−2ˆuˆv vˆ uˆ 0







 .

The additional nonlinear terms are given as NLT1 =0,

NLT2 = 1 Reρ²

2ˆuρ²_ˆy−2ρyˆ(ρˆu)yˆ+8

3uρˆ ²_xˆ− 8

3ρxˆ(ρˆu)xˆ+ 2 3vρˆ xˆρyˆ

−1

3ρxˆ(ρˆv)yˆ−1

3ρyˆ(ρˆv)xˆ

, NLT3 = 1

Reρ²

2ˆvρ²_xˆ−2ρxˆ(ρˆv)xˆ+8

3vρˆ ²_yˆ−8

3ρyˆ(ρˆv)yˆ+2 3uρˆ xˆρyˆ

−1

3ρxˆ(ρˆu)yˆ− 1

3ρyˆ(ρˆu)xˆ

, NLT4 = 1

Reρ²

3ˆu²+ 4ˆv²−4 γ

P rq+ 2c² (γ−1)P r

ρ²_yˆ+ 4ˆu γ P r −1

ρyˆ(ρˆu)yˆ

+ 1− γ

P r

(ρˆu)²_yˆ+ 4γ

P r − 16 3

ˆ

vρ_yˆ(ρˆv)_yˆ+ 4

3 − γ P r

(ρˆv)²_yˆ− 2γ P rρ_yˆE_yˆ +

3ˆv²+ 4ˆu²−4 γ

P rq+ 2c² (γ−1)P r

ρ²_xˆ+ 4ˆv γ

P r −1

ρˆx(ρˆv)ˆx

+ 1− γ

P r

(ρˆv)²_ˆx+ 4γ

P r − 16 3

ˆ

uρxˆ(ρˆu)xˆ+ 4

3− γ P r

(ρˆu)²_ˆx− 2γ P rρxˆExˆ

+ 2ˆuˆvρˆxρyˆ− 7

3ˆvρˆx(ρu)ˆ yˆ+ ˆuρxˆ(ρˆv)yˆ+ ˆvρyˆ(ρˆu)ˆx− 4

3(ρˆu)xˆ(ρˆv)yˆ

−7

3uρˆ yˆ(ρˆv)ˆx+ 2(ρˆu)yˆ(ρˆv)xˆ

.

We follow the idea of the one-dimensional case. Namely, we consider convection- and diffusion-dominated cases, and make a convex combination of them.

2.5.1 Convection-dominated case

For the convection-dominated case, we follow the idea in [31]. Assume the matrixA( ˆU) has four eigenvalues λ1 = ˆu−c, λ2 = λ3 = ˆu, λ4 = ˆu+c, and a complete set of left eigenvectors {l1,l₂,l₃,l₄} which forms the matrix

L=







 l₁ l₂ l₃ l₄









=









l1,1 l1,2 l1,3 l1,4

l2,1 l2,2 l2,3 l2,4

l_3,1 l_3,2 l_3,3 l_3,4 l4,1 l4,2 l4,3 l4,4







 .

We have LAL⁻¹ = Λ = diag(λ1, λ2, λ3, λ4) on the boundary. Let us assume λ1 < 0, λm ≥ 0, m = 2,3,4. With the characteristic decomposition V = LUˆ, the equation