Ching Y. Loh

Taitech, Inc., Beavercreek, Ohio

Philip C.E. Jorgenson

Glenn Research Center, Cleveland, Ohio

A Robust Absorbing Boundary Condition for

Compressible Flows

NASA/TM—2005-213654

October 2005

AIAA–2005–4716
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Ching Y. Loh

Taitech, Inc., Beavercreek, Ohio

Philip C.E. Jorgenson

Glenn Research Center, Cleveland, Ohio

A Robust Absorbing Boundary Condition for

Compressible Flows

NASA/TM—2005-213654

October 2005

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

17th Computational Fluid Dynamics Conference

sponsored by the American Institute of Aeronautics and Astronautics

Toronto, Ontario, Canada, June 6–9, 2005

AIAA–2005–4716
Acknowledgments

This work received support from the Quiet Aircraft Technology Project Office of NASA Glenn Research Center.

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7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

This report contains preliminary

findings, subject to revision as

analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov
Abstract

An absorbing non-reflecting boundary condition (NRBC) for practical computationsin fluid dynamics

and aeroacoustics is presented with theoretical proof. This paper is a continuation and improvement of

a previous paper [2] by the author. The absorbing NRBC technique is based on a first principle of nonreflecting, which contains the essential physics that a plane wave solution [1] of the Euler equations

remains intact across the boundary.

The technique is theoretically shown to work for a large class of finite volume approaches. When

combined with the hyperbolic conservation laws, the NRBC is simple, robust and truly multi-dimensional;

no additional implementation is needed except the prescribed physical boundary conditions. Several

numerical examples in multi-dimensional spaces using two different finite volume schemes are illustrated

to demonstrate its robustness in practical computations. Limitations and remedies of the technique are

also discussed.

1 Introduction

It is well-known that non-reflecting boundary conditions (NRBCs) play a key role in fluid dynamics and aeroacoustics computations and still remain an important topic in the current research areas. A spurious reflection resulting

from an inappropriate numerical boundary condition contaminates the flow field and may eventually spoil the entire

flow computation. Research on NRBC is a challenging topic in engineering and applied mathematics. A vast number

of papers on the topic of NRBCs have been published in past decades to treat various computational problems. Details

can be found in the recent reviews [3, 4, 5] and the references cited therein.

Generally there are two types of non-reflecting boundaries(see e.g., [3]). One (Type I) is the true domain boundary

where physical boundary conditions are specified and the other (Type II) is the artificially imposed boundary.

In this paper, we focus only on the Type I absorbing non-reflecting boundary. For a true boundary where the

actual physical boundary conditions are given as per the physical problem, the NRBC needs to play a dual role: on

the one hand it must be consistent with the given physical boundary conditions, on the other hand, it must be able to

absorb waves or disturbances from the domain interior.

For the absorbing NRBC, there are two ways to absorb the outgoing waves/disturbances: the ‘hard absorbing’ and

the ‘soft absorbing’. In the ‘hard absorbing’, the given boundary condition must be modified to an admissible one.

This can be done using the characteristics method. In one-dimensional flow, at the boundary, Enquist and Majda [6],

and Hedstrom [7] showed that a boundary condition considered to be non-reflecting is equivalent to saying that the

characteristic variables corresponding to the incoming characteristic curve remain unchanged across the boundary (see

NASA/TM—2005-213654 1

A Robust Absorbing Boundary Condition for Compressible Flows 

Ching Y. Loh 

Taitech Inc. 

Beavercreek, Ohio 45430 

Philip C.E. Jorgenson 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 
also Hirsch [8], p. 370). When the flow is subsonic, Riemann invariants (one-dimensional characteristic variables)

are used to turn a prescribed boundary condition into an admissible NRBC. For multi-dimensional flow, this onedimensional technique is combined with dimension splitting. Such a combined treatment has been the topic of many

papers (see e.g. [9] and references in [3]). The characteristics based NRBC is the most popular numerical treatment,

despite its complicated implementation.

The ‘soft absorbing’ treatments found in the literature generally include (but not limited to):

(i)A buffer zone (sponge zone) technique can be used to add a non-physical zone between the domain core and the

boundary to diminish the strength of the outgoing waves/disturbances before they reach the boundary, thus minimizing

the reflecting effect. In the buffer zone, either artificially increased dissipation is applied or grid stretching is imposed

(see e.g. [10, 18]), which is equivalent to numerical filtering.

(ii) In the recently developed perfectly matched layer (PML) method [11, 12] , a specially designed equation

system is imposed in the matching layer (or absorbing layer) at the boundary to guarantee the exponential decay of the

disturbances in the layer.

In addition to the existing absorbing NRBCs, as mentioned in [2], a new absorbing (Type I) NRBC technique was

recently discovered in an empirical way. The technique is robust but simple to implement and has been successfully

employed in many computations [16, 17, 18].

It is the purpose of the present paper to revisit the basic theoretical background of the absorbing boundary condition and to explain why the technique works well for flows in one to three dimensional spaces. Starting from a first

principle of non-reflection, i.e., there is no reflection at any interior point of a computational domain, a C

1

continuity

criterion for flow variables is introduced as a generic NRBC. The criterion is then incorporated in the time-dependent

hyperbolic conservation laws in discretized form, and the consequent absorbing (Type I) NRBC technique is described

and theoretically proven.

In the present paper, we first develop the basic principle and criterion for the continuous problem and then apply

them to the discretized numerical problems. The paper is arranged as follows: in §2, based on a first principle of

non-reflection, the C

1

continuity criterion of NRBC is introduced and proven. Based on this continuity criterion, the

absorbing NRBC is depicted in §3. Various numerical examples in two or three-dimensional spaces with two different

finite volume schemes are illustrated to demonstrate the capability of the new NRBC. Limitations and remedies of the

absorbing NRBC are discussed in §5. Finally, the paper is concluded with remarks in §6.

Throughout the present paper,

V = (v1, v2, v3, v4, v5)

T = (ρ, u, v, w, p)T

is employed to represent the primitive flow variables (for notations, see below). The primitive flow variables V and

the original non-conservative form of the Euler equations are conveniently employed in the theoretical investigation

but hyperbolic conservation laws in discretized form are needed in the numerical implementation. Here is the standard

three-dimensional conservation form of the non-dimensional Euler equations:

Ut + Fx + Gy + Hz = Q, (1)

where x, y, z and t are the streamwise and transverse coordinates and time, respectively. The conservative flow

variable vector, U, and the flux vectors, F, G, H, are given by:

U =





U1

U2

U3

U4

U5





=





ρ

ρu

ρv

ρw

ρe





,F =





ρu

ρu2 + p

ρuv

ρuw

ρuH





, G =





ρv

ρuv

ρv2 + p

ρvw

ρvH





, H =





ρw

ρwu

ρwv

ρw2 + p

ρwH





.

where u, v, w and ρ, p are respectively the velocity components, density and pressure, with the internal energy e =

p

ρ(γ−1) +1/2(u

2 +v2 +w2

), the enthalpy H = p/ρ+e, and γ = 1.4. The right hand side Q is the source term which

may possibly include the external forcing terms.

By considering (x, y, z,t) as coordinates of a four-dimensional Euclidean space, E4, and using the Gauss divergence theorem, it follows that Eq. (1) is equivalent to the following integral form of the conservation laws:

I

S(V )

Im · ds =

Z

V

QmdV, m = 1, 2, 3, 4, 5, (2)

NASA/TM—2005-213654 2
where S(V ) denotes the surface around a control volume, V , in E4 and Im = (Fm, Gm, Hm,Um) represents the flux

vectors. The vector differential ds = nds, where n is the outgoing unit normal vector of the infinitesimal surface

element ds in E4.

2 A first principle of non-reflection and the continuity criterion

In this section, a theoretical analysis of non-reflecting is conducted for the continuous problem (as opposed to a

discretized problem) of the Euler equations. Then, it is applied to the discretized NRBCs in the sections that follow. For

clarity, the flow variables are assumed to be continuous near the boundary, and all the spatial and temporal derivatives

exist. It should be stated that, even though discontinuitiessuch as shocks or contacts may develop and propagate across

the boundary, for a discretized absorbing NRBC, such continuous flow requirement is always satisfied locally at the

boundary via the numerical procedure, and hence the following first principle and criterion of non-reflecting are still

valid.

From the classical theory (e.g. Courant and Hilbert[1]) of partial differential equation (PDE), it is well known that

hyperbolic PDE systems support wave solutions. In the present paper, we specifically focus on the Euler equations.

Due to their hyperbolicity, they support linear continuous waves such as pressure pulse waves, entropy (density) waves,

and vorticity waves as well as nonlinear or discontinuous waves such as shocks.

2.1 A first principle of non-reflection

As the PDE system supports wave solutions, the next issue is focussed on the questions: what is ‘non-reflecting’

and how a boundary can become non-reflecting? Hereby, instead of following the conventional wisdom of a signal/wave propagating along characteristics, the concept of ‘non-reflecting’ is viewed from a different but simple perspective.

Recall a general observation that, no matter how crude the numerical scheme is, spurious reflection can only

be generated at the domain boundaries, no reflection ever occurs at any interior point. This is because only at the

boundary is the numerical solution subjected to two competing conditions, namely, the governing equations and the

imposed boundary conditons.

The following logical argument further justifies that no reflection should occur in the domain interior: At an

interior point of the domain, the solution is subjeted only to the governing equations. For any scheme that converges,

the numerical result at this point, which may even contain the possible spurious reflection propagated from a boundary,

is literally considered to be 100% a discretized solution of the governing equations and nothing else, disregarding any

discretization error. Conceptually, no such ‘impurity’ as ‘reflection’ is generated at this interior point.

Theoretically, for a hyperbolic PDE system, locally in a small neighborhood of any interior point O, by using

Cauchy’s method of the Fourier integral (e.g.,[1], p.210, or [ 8 ]), the homogeneous part of the solution of the initial

problem, which may contain oscillations, can be expressed by the superposition of plane waves:

V =

X

M

m=1

Z ∞

−∞

... Z ∞

−∞

V ̃ m(k)e

iθdk1dk2...dkn, (3)

Here, M is the number of PDE equations in the system, k = (k1, k2, ..., kn) is the n-dimensional wave number vector,

with kj , j = 1, 2, ..., n, ranging from −∞ to ∞, i =

√

−1, and x = (x1, x2, ..., xn) is the n-dimensional position

vector of this point O. V ̃ m = V ̃ m(k) and θ = θ(k, x,t) = x • k − ωmt are respectively the amplitude and the phase

of the plane wave, where ωm = ωm(k) (a real number) is the angular frequency and can be determined as a function

of the wave number vector k by substituting the single plane wave V ̃ e

iθ into the PDE system. For each plane wave

component in (3), as its amplitude and phase are functions of k and remain intact when passing through this interior

point O, no distortion or reflection occurs. So also for the solution V.

Based on such observation, logical reasoning, and the theory of hyperbolic PDE system, we conclude that if

all the points at the boundary are actually interior points of the domain, there is no reflection. A first principle of

non-reflection is then proposed and adopted in the present paper:

A boundary of the solution domain is said to be non-reflecting if every point on the boundary surface can be

considered as an interior point.

There are two requirements defining an interior point:

NASA/TM—2005-213654 3
t x

v

xo

D D

V V

i

i e

e

x

v

x

V

Vi

e

o

o

a b

 

C

C

1

0

t=t plane

slope

kink

Figure 1. a: Sketch of a one-dimensional scalar plane wave, b: the curves v = v(x,t) on a plane t = t0 at the boundary x = x0, showing

intuitively that C1

continuity is necessary for non-reflection.

1. the point must be physically located within the domain;

2. the consistency requirement that the governing PDE system is the only condition to be satisfied at this point.

Other conditions, if any, must be equivalent to or consistent with the PDE system.

In order that all the points along the boundary be interior points, of course, the solution domain must be extended

to include at least a small neighborhood around the boundary. Numerically, the ghost cell is a natural choice for the

neighborhood at the boundary. The consistency requirement is a more important issue. An NRBC is sought such that

it fully complies with the governing equations, that is, solutions in the domain interior and in the domain exterior must

be mutually a solution continuation of each other, satisfying the same PDE system in the extended domain.

This concept or definition of non-reflection leads naturally to a generic NRBC – the continuity criterion for Euler

equations, which will be proposed and proven in §2.2. The definition is a general concept and may be applied to

hyperbolic systems other than the Euler equations.

2.2 The continuity criterion - a generic NRBC

With the presence of the boundary ( a cylindrical hyper-surface), the four-dimensional Euclidean time-space E4 is

bisected into two portions, domain interior Di and domain exterior De (Figure 1a illustrates a 1-D scalar plane wave in

E2, only the real component of V is shown. The subscripts i and e stand respectively for domain interior and domain

exterior ). Within Di, the flow is governed by the original Euler equation system. It is also assumed that in the domain

exterior De, at least in a neighborhood of the boundary, the flow is still governed by an Euler equation system. If the

two Euler equation systems are identical at all the points of the boundary, the consistency requirement is satisfied, and

these points are interior points of the extended domain. The boundary is hence non-reflective.

The continuity criterion leads naturally to such non-reflection, it is stated as follows:

At each point of the boundary, the flow variables V and their spatial derivatives, namely, Vx, Vy, Vz, are

continuous; or, V is C

1

continuous in space across the boundary.

It can be easily shown that locally there is no reflection at the boundary surface if the continuity criterion is

satisfied.

Proof: Consider the non-conservation form of the Euler equations ( e.g., see [8]) for both domain interior and

exterior:

∂V

∂t

+ A ̃

∂V

∂x

+ B ̃

∂V

∂y

+ C ̃

∂V

∂z

= Q ̃ . (4)

Here Q ̃ is a nonhomogeneous source term vector and may be a vector function of x, y, z,t and V, and could include

forcing terms.

Note that the jacobian matrices A,  ̃ B ̃, and C ̃ are directly functions of V ( details may be found in [8]). As a result

of the C

1

spatial continuity of V, all the terms (items)

∂V

∂t

,

∂V

∂x

,

∂V

∂y

,

∂V

∂z

, A,  ̃ B ̃, C ̃, and Q ̃

in the governing Euler equations are identical across the boundary surface. Continuity of ∂V

∂t

is directly inferred

from (4). Now, (4) can be considered as a single governing equation system in an extended domain which covers

the boundary as its interior. The consistency requirement is satisfied at all the boundary points; hence, there is no

reflection at the boundary.

NASA/TM—2005-213654 4
2.3 Physical interpretation of the continuity criterion

For the three-dimensional Euler equations, corresponding to Eqn. (3), M = 5 and n = 3. As a special case of the

hyperbolic PDE system in §2.1, locally in a small neighborhood of an interior point, the homogeneous solution of the

Euler equations can always be decomposed into plane waves having a vector form [8]:

V = V ̃ e

iθ

, (5)

Here, as in §2.1, V ̃ = V ̃ (k, x, y, z) is the ‘amplitude’, and θ = θ(x,t) = x • k − ωt is the phase of the plane wave,

where k = (kx, ky, kz) is the given wave number vector and ω = ω(k) (a real number) is the angular frequency. and

x = (x, y, z) the position vector of the point O. Note that θ is a real number; θ = const. represents a space-time

characteristic surface or wave front [1].

Now, as the continuity criterion guarantees that a boundary point can be considered as an interior point, the plane

waves are employed in this paper to serve two purposes:

1. to demonstrate the physical effect of the continuity criterion: at such a boundary point, a plane wave, i.e., its

amplitude and phase remains intact or undistorted across the boundary. Hence the non-reflecting property is

justified as stated in §2.1;

2. to conduct phase error analysis at the boundary (in §5.1).

Figure 1b sketches how a simple one-dimensional scalar plane wave propagates across a boundary x = x0 in a

plane t = t0. The upper curve corresponds to C

1

continuity and exhibits no reflection. The lower curve corresponds

to C

0

continuity with a possible kink of slope or phase error, spurious reflection may occur.

The C

1

continuity criterion is a generic NRBC. When incorporated in certain finite volume approaches with the

hyperbolic conservation laws, a new class of absorbing NRBCs is created. As will be shown in §3 and §4, these

NRBCs are simple but robust. The continuity criterion may also have direct applications, but that would be the topic

of another NRBC paper.

2.4 Relation of the continuity criterion to the characteristics-based NRBCs

The continuity criterion is closely related to the well-known characteristics based NRBCs [6 - 9], although seemingly there is no apparent way to infer one from the other. First, both rely on the hyperbolicity of the equation system,

which supports wave/disturbance propagation and characteristics. Secondly, for the characteristics-based NRBCs, as

suggested in [6, 7], for the incoming characteristic variable wk, ∆wk should be set to zero across the domain boundary.

This step is consistent with the continuity criterion too. As such, their non-reflecting performance is expected to be

similar.

However, the continuity criterion treats a multi-dimensional plane wave as a physical entity and keeps it intact or

undistorted across the boundary surface to attain the ‘non-reflecting’ effect. This contrasts to the usual treatment of

characteristic field decomposition in the characteristics-based NRBCs and is much simpler.

3 The absorbing NRBC and the absorbing layer (matched layer)

The continuity criterion is primarily applied to the absorbing NRBC. This absorbing NRBC is generally considered to be a difficult problem because the boundary conditions are subject to excessive restrictions. On the one

hand, the given flow conditions must be satisfied; on the other hand, the boundary must be non-reflecting or absorbing

for the outgoing waves. Conventionally, by determining the one-dimensional outgoing or incoming characteristics,

the characteristics-based NRBCs modify the boundary conditions to an admissible one that handles the absorbing

boundary problem. The process is termed ‘hard absorbing’ in §1 and becomes cumbersome when the flow is multidimensional and complicated. The perfectly matched layer or PML method [11, 12] offers an attractive treatment,

in which a set of equations in the matched layer is prescribed and solved such that the outgoing wave is dissipated

exponentially and absorbed. Such absorbing occurs in the matched layer and hence belongs to the ‘soft absorbing’.

The new technique is shown to work for a large class of finite volume schemes that follow the two-stage procedure:

an evolution (time-marching) stage and a reconstruction stage. The continuity criterion is here combined with the

conservation laws in discretized form and yields a desirable absorbing NRBC - simple but robust. The non-reflecting

property of the Type I absorbing NRBC is established in §3.1. Section 3.2 provides a general guideline for numerical

implementation of the absorbing NRBC. The matched layer or absorbing layer is discussed in §3.3.

NASA/TM—2005-213654 5
















x

y

t

V

V

1

2

V=V +V 1 2

control volume

boundary surface element ∆S

a

O

centered at O

R L R

L

L

A R B

C

D

E F

O

∆S

boundary

surface

ghost cell

cntr.

ghost cell

b

A B

C

D

F E

O

ghostcell

center

boundary

c interior cell center

∆S

Figure 2. a: Boundary surface element ∆S as an internal interface of the control volume V ; b and c: Examples of 2-D numerical implementation:

b: A standard upwind scheme with an unstructured triangular grid; ‘R’ and ‘L’ represent respectively the right and left states for the Riemann solvers,

they are obtained by extrapolated flow data from the current cell center O and the neighboring cell centers D, E and F. The prism cylinder based

on ∆ABC is the space-time control volume. The boundary surface element ∆S practically coincides with the side AB. c: For the CE/SE method,

with AB as the boundary ∆S, the space-time control volume V is a hexagon cylinder based on ADBECFA. In both cases, physical boundary

conditions are given at the ghost cell center D.

3.1 The absorbing NRBC

After discretization, the boundary surface S is also divided into small boundary elements, ∆S, according to the

grid. Let V be a control volume which contains the boundary surface element ∆S as its internal interface (Fig. 2a).

Note that V may or may not be a grid cell. It will be shown that ∆S is non-reflective with any prescribed physical

boundary conditions at the ghost cell center. No additional numerical implementation is needed. The mechanism

works as follows: the C

1

continuity of the flow variables V across ∆S provides the non-reflecting property as shown

in the next paragraph. Then the inconsistent flow data between the imposed boundary conditions and the outgoing

wave/disturbance is treated as a ‘numerical jump’ around the boundary, captured, and absorbed by the shock-capturing

scheme which is based on the hyperbolic conservation laws. As a result, a matched layer (ML) or absorbing layer is

automatically generated around the boundary.

It is straightforward to prove the ‘non-reflecting’ property of the boundary surface element ∆S. In a finite volume

numerical approach, at the evolution stage (time-marching stage), for any prescribed physical boundary conditions at

the ghost cell center, the integral conservation laws (2) are solved with the control volume V , yielding a spatial average

V (or U) over V at the new time level n + 1. At the reconstruction stage, by using a flux limiter, such as the minmod

or the van Albada limiters etc., to recover the spatial gradients of V (i.e., Vx, Vy and Vz), this constant averaged V

is rebuilt into Vn+1 – a linear, high resolution (or second order accurate ) solution of (2) within the control volume

V . It is the reconstruction stage that provides the non-reflecting property. Note that Vn+1 represents a discretized

solution of (2) or a discretized ‘weak solution’ of (4) within the entire control volume V and that ∆S is an internal

interface of V . As a linear vector function within V , Vn+1 satisfies the C

1

continuity criterion at the surface center

of ∆S (Fig. 2a), hence ∆S is non-reflective.

3.2 Implementation of the absorbing boundary conditions

Construction of the absorbing NRBC is straightforward too – no additional numerical implementation is required

except the prescribed boundary conditions at the ghost cell centers, because an absorbing layer is automatically generated. For the class of finite volume approaches which contain the two-stage procedure of evolution (time-marching)

and reconstruction, here are some general guidelines:

1. Choose an appropriate control volume, V , at the boundary and ensure that the boundary surface element, ∆S, is

an internal interface of V . For most of the finite volume schemes, usually, a control volume V where the Gauss

divergence theorem (2) is applied is equivalent to a grid cell. Then, ∆S may coincide with the cell surface at the

NASA/TM—2005-213654 6
















modified to

admissible

NRBC

boundary

boundary

matched layer (ML)

outgoing

waves/

disturbances

outgoing

waves/

disturb.s

a b

given 

B.C.

given 

B.C.

Figure 3. Comparison of absorbing NRBCs at an inlet boundary; a: Conventional ‘hard’ absorbing NRBC (characteristics-based); b: Type I ‘soft’

absorbing NRBC, the matched layer plays the role as a buffer layer between the inconsistent flow data at domain boundary and interior.

domain boundary but is considered lying at the inner side of the boundary surface (Fig. 2b). For some schemes,

e.g. the CE/SE scheme [14,15], a control volume V is larger than a grid cell and ∆S is truly an internal interface

of V (Fig. 2c).

2. The finite volume approach follows the standard two-stage procedure: the evolution (time-marching) stage and

the reconstruction stage by flux limiter. Nevertheless, even when the reconstruction stage is missing, the NRBC

still holds, as the case is equivalent to a first order scheme, with all the spatial gradients vanished.

This absorbing NRBC is flexible and valid with practically any cell shapes, structured or unstructured. Figure 2b

and 2c show examples of the two-dimensional numerical implementation – the treatments in a standard upwind scheme

and in the CE/SE method. Both will be used as numerical platforms for the computed examples. Flow data at the

boundary cell center O is updated based on flow data at its three neighboring cell centers D, E and F. The divergence

theorem is applied to the control volumes: the triangle cylinder ∆ABC or the hexagon cylinder ADBECF. The side

AB is the non-reflecting boundary and an internal interface of the control volume as well.

3.3 The matched layer (absorbing layer) near the boundary

As the specified boundary conditions are imposed at the ghost cell centers, they also play a dual role of an

absorbing (Type I) NRBC. Often, after a considerable long run-time, inconsistency between the imposed boundary

conditions and the flow in the domain interior near the boundary may develop. Then, while the C

1

continuity across the

boundary surface rules out spurious reflections, the shock-capturing scheme automatically captures the inconsistency

in a ‘matched layer’ (ML) or ‘absorbing layer’. Decaying of the inconsistency in the absorbing layer is fast. Depending

on the required accuracy, the matched layer (ML) is typically only a few cells thick. It is non-physical, but plays an

important role in separating and matching the inconsistent flow data at the boundary and the domain interior. The

matching or absorbing is thus considered ‘soft’.

The situation is similar to that in the PML (perfectly matched layer) method [11, 12], except that the ‘matched

layer’ or absorbing layer is automatically generated and no special equations in the matched layer are needed.

4 Numerical examples

The absorbing NRBC is the primary but indirect application of the continuity criterion. As a matched layer or

absorbing layer will be formed automatically, the absorbing NRBC is flexible and capable of handling complicated

non-reflecting situations, such as with the ‘difficult’ cases where locally the flow disturbance may enter or exit the

domain through the boundary and where the flow conditions could be supersonic along a portion of the boundary but

subsonic along another portion of the same boundary.

Since computations in multi-dimensional space are of primary importance, in this section, the absorbing NRBC

is tested in examples in two- or three-dimensional space with different finite volume schemes to demonstrate its

performance and robustness. Here, a standard upwind scheme with an unstructured triangular grid, as depicted in

Fig. 2b (for 2-D problems) and the space-time conservation element and solution element (CE/SE) method [13, 14]

are chosen as the numerical platforms for computing the examples. The upwind scheme uses Roe’s approximate

Riemann solver and renders first order accuracy in time but high-resolution in space. The CE/SE scheme is nominally

NASA/TM—2005-213654 7

second order accurate in both space and time.
pulse

t=6 t=30

vorticity/

entropy

wave

t=60

Figure 4. Propagation of a linear acoustic pulse with vorticity/entropy disturbances: density contours plots at t = 6, 30, 60; with comparison to

exact solution along the x axis, showing no reflection at the outflow boundary. The CE/SE scheme is employed here.

1 2 3 4

5 6 7 8

9 10 11 12

 

Figure 5. Plots 1-12, propagation of a Gaussian pulse in a circular domain from t = 60 to t = 120. The flow Mach number, M, is 0.2 at an

angle of 45o. Although the flow is inclined and the boundary is curved, no visible reflection is observed. A first order Godunov upwind scheme is

employed in this problem. The absorbing layer exists but probably too thin and weak to be visible.

4.1 2-D acoustic pulse, entropy wave, and vorticity wave propagation

This problem illustrates the propagation of the three basic types of linear waves in a two-dimensional domain [15,

16]. The computational domain in the x-y plane is a square with −100 6 x 6 100, and −100 6 y 6 100. A uniform

NASA/TM—2005-213654 8
1 2 3

4 5 6

Figure 6. Plots 1-6, propagation of a Gaussian pulse in an egg-like domain at t = 150, 200, 250, 300, 350, and 400. The flow Mach number,

M, is 0.05 and directs at 45o. ( about 65,000 unstructured triangular cells, with the CE/SE scheme)

grid with 40, 000 uniform triangulated cells is used with the CE/SE scheme. Initially, a Gaussian acoustic pulse is

located at the center of the domain (0,0) and a weaker entropy/vorticity disturbance is located off the center at (67,0),

with a mean flow Mach number, M = 0.5, in the x direction. The corresponding non-dimensional pressure, density,

and streamwise and transverse velocity components are given by

p =

1

γ

+ δe

−α1(x

2+y2

)

, ρ = 1 + δe

−α1(x

2+y2

) + 0.1δe−α2[(x−67)2+y

2

]

,

u = M + 0.04δye

−α2[(x−67)2+y

2

]

, v = −0.04δ(x − 67)e

−α2[(x−67)2+y

2

]

,

where α1 = ln 2/9, α2 = ln 2/25, and with a small amplitude factor, δ = 0.001, the two-dimensional Euler equations

are practically linearized. At all four boundaries, flow variables are given with the M = 0.5 mean flow conditions:

pa =

1

γ

, ρ = 1, u = M = 0.5, v = 0.

These conditions play the role as an absorbing NRBC. Figure 4 illustrates the evolution of the density contours at

different time steps with comparison to the exact solutions (perturbation part) along the x axis. It is observed that the

numerical solutions agree well with the exact solutions and that there is no reflection along x axis. As the the waves

are weak, the absorbing layer is not visible.

4.2 2-D acoustic pulse propagation

The second example is a two-dimensional subsonic non-linear wave propagation problem. A circular computational domain in the x-y plane is used, ranging from −100 6 x 6 100, and −100 6 y 6 100 (Fig. 5). There are about

87,200 unstructured triangular cells in the domain, and the two-dimensional unsteady Euler equation system is solved

by a simple first order accurate Godunov upwind scheme. Initially, a Gaussian acoustic pulse is located at (10, 10) in

the domain, with a mean flow Mach number, M, of 0.2 in a direction 45oinclined to the x axis.

p = e

−ln2(x

2+y2

)/9 + 1/1.4, ρ = e−ln2(x

2+y2

)/9 + 1, u = v = 0.1

√

2.

The Type I NRBC which is identical to the initial conditions i.e. p = 1/1.4, ρ = 1, u = v = 0.1

√

2, is imposed at

the circular boundary. As time elapses, the non-linear pulse convects with the flow and propagates. Figure 5 shows

twelve frames of the isobars from t = 65 to t = 120 at an increment of ∆t = 5. When the pulse propagates towards

the boundary and exits the domain, no visible reflection is observed. The matched layer is too thin to be visualized.

The curved boundary and the oblique flow direction help to demonstrate the capability of the absorbing NRBC.

NASA/TM—2005-213654 9
1 2 3

4 5

p:

0.01 −0.75

shock

expansions

(a)

1 2 3

4 5

rho:

.01 − 1.5

shock contact

contact

expansions

(b)

Figure 7. Two-dimensional radial Riemann problem, showing how a shock and a contact discontinuity are perfectly absorbed by the boundary.

Plot 1-5 correspond to their propagation at t = 50, 100, 125, 175, and 225. The mean flow is stationary (M = 0).

4.3 2-D acoustic pulse propagation in an egg-like domain

This example demonstrates that the absorbing NRBC can be applied to a domain of more general shape. All the

conditions are similar to the previous example except that the domain has the shape of an egg and that the mean flow

Mach number, M, is 0.05. The numerical scheme used here is the CE/SE scheme. Figure 6 shows six frames of the

isobars from t = 150 to t = 400 at an increment of ∆t = 50. In this case, when the pulse propagates towards the

boundary and exits the domain, the absorbing NRBC performs generally well, except some minor reflection in the last

3 plots, due to certain phase errors that will be described in §5.1.

4.4 Propagation of discontinuous waves (1)

Here the problem is a two-dimensional radial Riemann problem and the physical phenomena are complicated.

This example demonstrates with the CE/SE scheme how a shock and a contact discontinuity are absorbed by the

boundary. The same circular domain as in §4.2 is employed. The mean flow is stationary: (ρ0, u0, v0, p0) =

(0.5, 0, 0, 0.1/1.4), which are also imposed at the ghost cell centers at the circular boundary as the absorbing NRBC.

Initially, within an inner circle of radius r = 50, a state of high pressure and density is given: (ρ, u, v, p) =

(1.0, 0, 0, 1.0/1.4). Figure 7 shows five plots of isobars and isopycnics (density contours) at different time steps.

In Fig. 7a, the first two plots show how the circular shock approaches the absorbing boundary. From the third plot on,

the shock passes through the boundary and is absorbed by the boundary, leaving no trace of reflection. Figure 7b not

only demonstrates how the shock leaves the domain, but also the evolution of the contact discontinuity. In the fifth

plot, the contact already passes through the boundary and no reflection appears.

4.5 Propagation of discontinuous waves (2)

This radial Riemann problem is similar to the previous example, except that the mean and core flows move at M =

0.2 and 45o. Initially, the mean flow and core flow data are respectively (ρ0, u0, v0, p0) = (0.5, 0.1414, 0.1414, 0.1/1.4),

and (ρi, ui, vi, pi) = (1.0, 0.1414, 0.1414, 1.0/1.4). Figure 8 shows eight plots of isopycnics (density contours) at different time steps. The circle-like shock moves towards the northeast direction in plots 1 and 2. In plots 3 and 4, the

shock is almost fully absorbed by the boundary, leaving no visible reflections. From plot 5 on, the contact propagates

slowly towards the boundary and is well absorbed. A high-resolution upwind scheme with a minmod flux limiter is

employed for computing this case.

4.6 2-D subsonic cavity flow

Figure 9 illustrates the instantaneous isobars for a cavity noise problem. The grid consists of 42, 000 triangulated

structured cells. The problem is a M = 0.8 flow past a rectangular cavity of aspect ratio of 6.5. At the cavity walls,

no slip boundary condition is imposed. Due to vortex shedding and acoustic feedback at the cavity edges, strong

nonlinear acoustic waves are generated and propagate in both upstream and downstream directions [17]. Figure 10

NASA/TM—2005-213654 10
1 2 3 4

5 6 7 8

shock contact

contact expn.

Figure 8. Two-dimensional Riemann problem with mean and core flows moving at 45O and M = 0.2, showing how the the shock and contact

discontinuity are absorbed at the boundary. Plots 1-8 correspond to the density contours at t = 40, 60, 80, 90, 100, 120, 140, and 160.



D

3D

6.5D

M=0.8

Figure 9. Isobars snapshot for a cavity noise problem (Mach number M = 0.8), showing inflow NRBC and its absorbing property. Also, no

visible reflection is found at the top and the outflow boundary

is an enlargement of Fig. 9 around the inflow area, revealing the details of the contours at the matched layer. It is

observed that there is no spurious reflection and that the matched layer (absorbing layer) is about 4-5 cells thick. A

CE/SE type Navier-Stokes solver is used in the computation.

4.7 3-D circular jet screech problem

The absorbing NRBC can be extended to three-dimensional space problems in a consistent way. Figure 11 is the

sketch of an underexpanded circular jet in three-dimensional space. The jet nozzle protrudes into the computational

domain by S = 2D, D being the inner diameter of the nozzle. The unstructured mesh contains 3.7 million hexahedral

cells. At the inlet plane, an ambient (stationary) condition or the Type I absorbing NRBC is specified. Jet flow at

higher pressure (with jet Mach number Mj = 1.42) is specified at the nozzle exit. For over or under expanded jets,

NASA/TM—2005-213654 11
matched

layer

acoustic 

wave

cavity

Figure 10. Details of the contours at the inflow boundary, showing the matched layer (absorbing layer) at the inlet and its spreading over the grid.

L

2R

S

inner

diam. D

mid−plane

screech wave

isosurfaces

matched

layers at

inlet

Figure 11. Left: Sketch of the 3-D circular jet screech problem and computational domain, jet Mach number Mj = 1.42, S = 2D, L = 10D

and R = 5.5D. Right: instantaneous pressure isosurfaces, showing screech waves, and in particular, the disc of matched layer at the ambient inlet.

screech tone noise and waves may occur in a feedback loop in which the instability waves (vortices) in the shear

layer interact with the shock-cell structure in the jet core and generate screech waves that propagate upstream to the

nozzle lip triggering new vortices. A CE/SE type three-dimensional Navier-Stokes solver is employed for the work.

Figure 11 also shows a snapshot of the pressure isosurfaces after running 510,000 time steps. The helical screech

waves are clearly illustrated. At the inlet, an absorbing layer (or ML) is automatically formed as shown in Fig. 11.

5 Discussions on the NRBCs

Generally, the new NRBC exhibits little reflection. However, due to discretization errors and the possible consequent phase error at the physical wave front, there are situations when the NRBC may fail and discernible reflections

occur. Limitations and remedies are briefly discussed here.

5.1 Limitation of the NRBCs

Due to discretization and the possible consequent phase error, the accuracy of the NRBC could be degraded.

Consider the Fourier mode in a plane wave solution (5), i.e., e

i(k•x−ωt)

. Here, θ(x,t) = k • x − ωt is the phase of

NASA/TM—2005-213654 12
boundary

surface

wave front (characteristic surface)

a b

Figure 12. Wave propagation across a boundary; a: wave front coincides with boundary surface, best non-reflecting effect is obtained; b: wave

front is inclined to boundary surface, phase error may occur and cause spurious reflection. Arrows indicate wave propagation direction (direction

of wave number vector)

Figure 13. Mach radiation from a M = 2 axisymmetric jet; Top: no buffer zone is added, flow field is contaminated; Bottom: with buffer zone

added (not shown), showing a clean acoustic field.

the wave mode, with k being the given wave number vector in the propagation direction, and ω the frequency. The

characteristic surface, θ(x,t) = const., stands for a wavefront ( see e.g., Courant and Hilbert [1] or Hirsch [8]). After

discretization, the center, O, of ∆S (Fig. 2a) is employed to represent the entire ∆S (a hyper-surface in both time and

space), both spatial and temporal phase errors may occur. While the temporal phase error is part of the discretization

error, the spatial phase error is closely related to the spurious reflections and is less tolerable. Assume that time t is

held unchanged, the spatial phase error can be easily analysed: Let x = (x, y, z) be the position vector of any point

on ∆S and xO the position vector of the center O. Then, the spatial phase error ∆θ due to replacing x by xO is:

∆θ = k • x − k • xO = k • (x − xO) (6)

note that the vector ∆x = x − xO lies on ∆S, and ∆θ = 0 when k is normal to ∆S. Therefore, spurious reflection

is minimized when the wave front (characteristic surface) coincides with the boundary (Fig. 12a). Otherwise, a phase

error of order O(∆x) may be introduced and cause numerical reflection. It should be noted that the above analysis

involves only discretization and is hence valid for other NRBC techniques such as the characteristics-based NRBC.

It is conceivable that changing the orientation of the boundary will help to reduce spuriousreflections. But in practice, an effective remedy is to impose a buffer/sponge zone between the boundary and the domain interior. Although

the same governing equations (1) or (2) are employed in the sponge zone, numerical damping is highly increased

to diminish the wave/disturbance amplitude before it reaches the boundary and to minimize the spurious reflection

typically by grid stretching. An example is illustrated here to demonstrate the effectiveness of the buffer/sponge zone.

NASA/TM—2005-213654 13
5.2 Application of the buffer/sponge zone

Consider a two-dimensional axisymmetric jet with Mach radiation. The domain size is 33D × 19D with D being

the jet nozzle diameter. A grid of 300 × 280 non-uniform rectangular cells is used and then triangulated. Initially, a

Mach number M = 2 jet exists. At the center of the nozzle exit plane, a source is imposed and perturbs the jet flow

with a small amplitude of 0.001 at a Strouhal number St = 0.2. As a result of the local supersonic phase velocity

at the outskirts of the shear layer, Mach radiation is triggered and gains its strength along the stream. At the outflow

boundary, particularly at the shear layer, due to the strong Mach waves with inclined angle, the absorbing outflow

NRBC fails and spurious reflections are generated and propagate upstream (Fig. 13, left). After a buffer zone of 10

cells is added at the domain outflow boundary, the spurious reflections disappear and a clean Mach wave radiation is

shown (Fig. 13, right plot). The size of the buffer zone cells stretches exponentially at a rate of 20% along the x axis

and thus highly enhances the numerical dissipation or filtering effect in the scheme.

6 Concluding Remarks

In the present paper, the problem of non-reflecting boundary condition is reviewed from a different perspective and

a first principle of non-reflection, that no reflection occurs at any interior point of the domain, is proposed. A generic

NRBC - the C

1

continuity criterion is then derived. The criterion contains the essential physics that the the plane wave

solutions are treated as physical entities and remain intact when propagating across a non-reflecting boundary surface.

Such a criterion may also be applied more generally to hyperbolic systems other than the Euler equations.

The criterion leads to a new class of NRBCs. A primary outcome is the Type I absorbing NRBC, which is

designated for a true boundary with prescribed physical boundary conditions. The technique works well with a large

class of finite volume approaches that follow the two-stage procedure of evolution and reconstruction. The absorbing

NRBC is simple, robust and truly multi-dimensional, no additional numerical implementation is needed. It is valid

for structured or unstructured grid with any cell shape, disregard the flow is supersonic or subsonic. No dimension

splitting is exercised and no complicated characteristic variable or Riemann invariant decomposition is required. The

absorbing NRBC often leads to automatic generation of a matched layer or absorbing layer with a stencil of a few cells.

This matched layer is similar to the well-known PML (perfectly matched layer) in [11, 12] but is formed automatically

with no additional treatment required.

A limitation of the NRBC and the remedy is also discussed. If the wave propagation direction is inclined to the

boundary, the NRBC may perform poorly due to inherent phase errors (§5.1). Sometimes, a combination of the NRBC

treatments may provide much improved results, such as the incorporation of a buffer zone.

In practical computations with the Navier-Stokes equations for flows at high Reynolds numbers, as the continuity criterion is still valid and the viscous terms only introduces small variations, the absorbing NRBC may still be

employed with satisfactory results, as observed in the examples in §4.6 and §4.7.

The absorbing NRBC works well with discontinuous waves, as demonstrated in the examples in §4.4 and §4.5,

since the continuity criterion is still in effect within the control volume at the boundary after the the reconstruction

stage (§3.1).

Applications of the C

1

continuity criterion is not limited to the absorbing NRBC, it may also be directly applied

to an artificial boundary and form the Type II NRBC, which will be the topic of another paper.

References

[1] Courant, R. and Hilbert,D. “ Methods of Mathematical Physics”, Vol.II, John Wiley & Sons, London, 1962.

[2] Ching Y. Loh, “ On a Non-Reflecting Boundary Condition for Hyperbolic Conservation Laws”, AIAA Paper

2003-3975 (2003).

[3] Givoli,D., “Non-reflecting Boundary Conditions”, J. Comput. Phys., Vol. 94, 1-29(1991).

NASA/TM—2005-213654 14
[4] Tam, C.K.W., “Advances in Numerical Boundary Conditions for Computational Aeroacoustics”, J. Comput.

Acoust., Vol. 6, 377-402(1998).

[5] Hu, F.Q., “Absorbing Boundary Conditions”, Intl. J. Comput. Fluid Dynamics, Vol. 18(6), 513-522(2004).

[6] Engquist, B. and Majda, A., “Absorbing boundary conditions for the numerical simulation of waves”, Math.

Comp., Vol. 31, pp. 629-651 (1977).

[7] Hedstrom,G.W., “ Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems”, J. Comput. Phys. ,

Vol.30, 222-237 (1979).

[8] Hirsch, C. “Numerical Computation of Internal and External Flows” Vol. I and II, John Wiley & Sons (1993).

[9] Thompson, K. W. “Time-dependent boundary conditions for hyperbolic systems”, i & ii, J. Comput. Phys., Vol.

68, pp. 1-24, (1987); also Vol 89, pp 439-461, (1990).

[10] Rai, M.M. and Moin, P., “Direct Numerical Simulation of transition and Turbulence in a Spatially Evolving

Boundary Layer”, AIAA Paper, 95-0164 (1995).

[11] Hu, F.Q., “On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer”,

J. Comput. Phys., Vol.129, 201-219 (1996).

[12] Hesthaven, J.S.,“The Analysis and Construction of Perfectly Matched Layers for the Linearized Euler Equations”, ICASE Report No. 97-49 (1997) (also in J. Comput. Phys.)

[13] Chang, S. C., Wang, X. Y. and Chow, C. Y., “ The Space-Time Conservation Element and Solution Element

Method – A New High-Resolution and Genuine Multidimensional Paradigm for Solving Conservation Laws ”,

J. Comput. Phys. vol. 156, pp. 89-136 (1999).

[14] Wang, X.-Y. and Chang S.-C., “ A 2-D Non-splitting Unstructured Triangular Mesh Euler Solver Based on the

Space-Time Conservation Element and Solution Element Method” CFD J. vol. 8, pp309-325 (1999).

[15] “ICASE/LaRC Workshop on Benchmark Problemsin Computational Aeroacoustics”, Eds. J.Hardin,J.R.Ristorcelli,

and C.K.W.Tam, NASA/CP-3300 (1994).

[16] Loh, C. Y., Hultgren, L. S. and Chang S.-C., “Computing Waves in Compressible Flow Using the Space-Time

Conservation Element Solution Element Method,” AIAA J., Vol. 39, pp. 794-801 (2001).

[17] Loh, Ching Y., Wang, Xiao Y., Chang, Sin-Chung and Jorgenson, Philip C.E., “Computation of Feedback

Aeroacoustic System by the CE/SE Method”, in proceedings of the 1st Int’l Conf. on CFD (ICCFD), Kyoto,

Japan, July, 2000; (published by Springer-Verlag, 2001).

[18] Loh, C. Y. and Zaman,K.B.M.Q., “Numerical Investigation of Transonic Resonance with a Convergent-Divergent

Nozzle”, AIAA J., Vol. 40, No. 12, pp. 2393-2401(2002).

NASA/TM—2005-213654 15
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AIAA–2005–4716

E–15143

WBS–22–781–30–69

21

A Robust Absorbing Boundary Condition for Compressible Flows

Ching Y. Loh and Philip C.E. Jorgenson

Non-reflecting boundary condition; Compressible flow

Unclassified -Unlimited

Subject Categories: 02 and 64

Prepared for the 17th Computational Fluid Dynamics Conference sponsored by the American Institute of Aeronautics

and Astronautics, Toronto, Ontario, Canada, June 6–9, 2005. Ching Y. Loh, Taitech, Inc., 1430 Oak Court, Suite 301,

Beavercreek, Ohio 45430, Ching.Y.Loh@grc.nasa.gov.; and Philip C.E. Jorgenson, NASA Glenn Research Center,

jorgenson@nasa.gov. Responsible person, Ching Y. Loh, organization code RTS, 216–433–3981.

An absorbing non-reflecting boundary condition (NRBC) for practical computations in fluid dynamics and aeroacoustics

is presented with theoretical proof. This paper is a continuation and improvement of a previous paper by the author. The

absorbing NRBC technique is based on a first principle of non-reflecting, which contains the essential physics that a

plane wave solution (Fourier mode) of the Euler equations remains intact across the boundary. When combined with the

hyperbolic conservation laws, the NRBC is simple and robust, provided the numerical scheme maintains locally a

solution of second order (or higher) accuracy near the boundary. Several numerical examples in multi-dimensional

spaces are illustrated to demonstrate its robustness in practical computations.