_r__ ;!1,

NASA Contractor Report 182048

ICASE Report No. 90-37

ICASE

STABLE BOUNDARY CONDITIONS FOR CARTESIAN

GRID CALCULATIONS

M. J. Berger

R. J. LeVeque

Contract No. NAS1-18605

May 1990

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, Virginia 23665-5225

Operated by the Universities Space Research Association

FOR C_._TESiAN

(TCASE) ]7 p

(NAgA-CR-I_204d) STA_LE BOUNDARY CONdiTIONS

GRID CALCULATIONS Final Report

C_CL 12A

G3/_4

NASA

National Aeronaulics and

Space Adminislralion

N90-23139

Unclas

0281015

Langley Research Center

Hamplon, Virginia 23665-5225

Stable Boundary Conditions for Cartesian Grid Calculations

M. J. Berger 1

Courant Institute

251 Mercer St.

New York University

New York, NY 10012

and

R. J. LeVeque 1

Department of Mathematics

University of Washington

Seattle, WA 98195

ABSTRACT

We solve the inviscid Euler equations in complicated geometries using a Cartesian

grid. This requires solid wall boundary conditions in the irregular grid cells near the

boundary. Since these cells may be orders of magnitude smaller than the regular grid

cells, stability is a primary concern. We present a new approach to this problem and

illustrate its use.

IThe authorswere supportedinpartby NSF GrantsASC-8858101 and DMS-8657319,AFOSR

Grant 86-0148,and DOE Grant DE-FG02-88ER25053. This work was alsosupportedinpartby

the NationalAeronauticsand SpaceAdministrationunderNASA ContractNo. NAS1-18605 while

theauthorswereinresidenceattheInstituteforComputer ApplicationsinScienceand Engineering

(ICASE),NASA LangleyResearchCenter,Hampton, VA 23665.

1 Introduction

In previous work [1], [6], [7], we have described a Cartesian grid method for the

inviscid Euler equations in arbitrary geometries. There are many advantages to be

gained from this approach. Grid generation is simplified, since we avoid the use of

(possibly multiblock) body-fitted grids, and we can use high resolution, highly efficient

solvers on regular grids over the bulk of the domain. This has led to renewed interest

in Cartesian grids in recent years, e.g., [3], [10]. One of the difficulties with Cartesian

grids is that they give insufficient resolution in certain regions such as leading edges.

This can now be overcome by Cartesian adaptive mesh refinement [1], [2].

The principal remaining difficulty in this approach is due to the essentially arbitrary way that a Cartesian grid intersects the boundaries of the computational

domain. In particular, a solid wall boundary cutting through the grid creates irregular cells that may be orders of magnitude smaller than the regular cells away from

the boundary. For these irregular cells, special difference equations are needed that

maintain stability and accuracy, and satisfy the solid wall boundary conditions of no

normal flow.

In this work, we present an improved method for the small boundary cells. We

use an explicit, finite volume formulation that computes fluxes at cell edges on the

regular part of the domain. We would like to define fluxes at the edges of the irregular

cells in such a way that the method is stable even with very small boundary cells,

using a time step based on the regular grid ceils away from the boundary. The CFL

condition requires that the numerical method allow information to propagate at least

as quickly as the underlying differential equation. In the present context this means

that we must define fluxes at the sides of our irregular cells based on more than just

the neighboring cell values.

In our previous work, we have used a wave propagation approach in defining these

fluxes. Here we propose an alternative method that has some advantages over the

wave propagation approach. In particular, the wave propagation method is subject

to intermittent instabilities due to two-dimensional effects that are not clearly understood. The new method has a cancellation property in two dimensions that appears to

give better stability properties. Moreover, the computational geometry is simplified

in the new approach. The fluxes are defined in terms of weighted averages of nearby

cell values. These weights may be calculated as a preprocessing step on any fixed grid

and need not be repeatedly calculated. In the previous approach the weights depend

on the flow variables and a certain amount of computational geometry was required
nearthe boundary in every time step.

Weconsiderthe inviscid Euler equationsof gasdynamicsin two spacedimensions,

_, + f(_). + g(_)_= 0 (1.1)

where

l

,_= P_" :(_)= _'_+ P g(_,)= /'_:" (1.2)

pU2 ' pUl U2 ' pu_ + p

pZ ul(pE + p) u2(pZ + p)

Here (ul, u2) represents the velocity, E is the total energy per unit mass, and p is the

pressure, which is related to the other variables by the equation of state. We assume

a "7-law gas, so that

1 2

p = ("7 - 1)(pE - _p(u I + u_)). (1.3)

At a solid wall boundary we require that the component of velocity normal to the

wall be zero.

In one space dimension the system reduces to

u, + f(u)x = 0 (1.4)

where u = (p, pv, pE) and f(u) = (pv, pv 2 + p,v(pE + p)), with v = ul the velocity.

The boundary conditions become v = 0 at a solid wall.

2 A one-dimensional example

In order to illustrate this approach we begin with a 0he:dimensional model problem,

the one-dimensional Euler equations for x > 0 with a solid wail at x = 0. We take a

grid with cell interfaces at the points

mo----O

X 1 _--- h #

x i = h I + jh for j = 2, 3, ....

Here h is a uniform grid spacing and h _ < h. the grid is uniform except for one small

cell near the boundary (see Figure 1). We use a conservative method in the form

k ,,

U_'+1 = U_/- _-_/[r;+ 1 - F_/], j = 0, 1, .... (2.5)

2
I

21

zO=O

1

_2

I

_3

Figure 1: One dimensional grid with one irregular cell adjacent to the wall.

Here hj is the width of the jth cell, so in our case we have h0 = h' and hj = h for

j>0.

For simplicity we restrict our attention to Godunov's method on the regular portion of the grid, although the ideas we propose can be extended to higher order

methods as well. In Godunov's method we take

,F_j-"f (u*(U_j_I,U'_)) (2.6)

where u'(u _, z_n) represents the solution to the Pdemann problem with left and right

states u _ and u s, evaluated along z/t = O. Although a rigorous stability proof is not

available for systems of equations, in practice this method is always stable provided

the CFL condition

is satisfied, where A,,,,, is the maximum wave speed. We will assume that our time

step k is chosen so that the condition (2.7) is satisfied relative to the uniform h. We

will use the flux (2.6) for j = 2, 3, ..., i.e., at all interfaces where the cell on both

sides is regular. Our task is to define fluxes F_ for j = 0, 1 so that we maintain

stability (and accuracy) with this time step even if h' << h.

First suppose h' = h. Then we can use the Godunov flux (2.6) also at j = 1.

At the wall we use the well-known observation that the solution to the boundary

value problem can be obtained by ignoring the wall and extending the computational

domain to the whole line -c_ < z < o0, if we take data zt0(z) for z < 0 equal to

/,(=,o) = p(-=,o)

v(=,o) = -,.,(-.--,o)

p(=,o) = p(-=,o).

forz <0

We will denote this "reflection" of the data (in which the velocity is negated) by the

operator 7_, so that for shorthand we can write

_,(=,o)= _(,.,(-=,o)) for=< o.
With this extended data, the solution continues to satisfy u(x, t) = T_(u(-x, t)) also

for t > 0 and in particular the boundary condition u(0,_) = 0 is automatically

satisfied. This suggests that we obtain a flux at the wall by solving a Riemann

problem with left and right states

= 7e(v0), u0

in each time step to obtain

Fo = f(u*(Tt(Uo), Uo).

(For brevity we will leave off the superscript n in general.) Note that the density

and energy components of this flux will be zero since the velocity component of u* is

zero at the wail. There will only be a momentum flux at the wall due to the pressure

there, as expected physically.

If h _ <: h we could attempt to use this same formula to define F0 but we would

find that it is unstable unless the CFL condition

is satisfied. This will place an unreasonable restriction on k if h' << h.

This instability is caused by the fact that the boundary flux F0 is based on the data

U0 alone. If the CFL condition (2.8) is satisfied, then it is only this data that affects

the flux at the wall over the time step. However, when (2.8) is violated the value Ux

should also affect the flux at the wall, and ignoring this effect leads to instability.

In a "large time step" approach we increase the stencil of the method, meaning we

allow more data points to come into the computation of each flux, and hence retain

stability. One way to achieve this is by a wave propagation approach. The solution of

the Riemann problem at each cell interface consists of three waves propagating away

from the interface. If (2.8) is satisfied then these waves remain in the cells bordering

the_nterface during the entire time step and hence affect the solution only in these

cells. If (2.8) is violated then the waves may affect cells further away. Implementing

Godunov's method in terms of this wave prppagation approach, allowing waves to

affect more than just the adjacent cell, gives a large time step generalization that

remains stable for much larger time steps[5]. In the present context this allows us

to reduce h' without reducing the time step k. Waves from the bOundary Riemann

problem cross the interface at 0_1and affect U1 as well as U0. Waves from the interface

at xl may reach the boundary. These waves reflect off the boundary and the reflected

4
wave affectsthe value U0 and perhaps also U1 if the reflected wave reaches the cell

interface at zl during the time step.

A more detailed description of this procedure may be found in [7]. A natural

extension to two space dimensions gives one method to deal with small cells near

the boundary, as described in [1], [61, [7]. In one dimension this works very well

but in two dimensions occasional stability problems have still been observed due to

multidimensional effects.

The new approach. Our new approach to the small cell problem can also be

illustrated with the one-dimensional problem described above. We again use the

method (2.5) with Godunov fluxes (2.6) for j = 2, 3, .... For j = 0 and j = 1 we

define fluxes in a similar manner but with a different choice of states ur and u s in the

Riemann solver. Recall that in a naive attempt to use Godunov's method regardless

of the size h' of the small cell we would take left and right states

R

u_ = _(Uo) Uo = Uo (2.9)

(2.1o)

To maintain stability when h' is small, we need to allow data from additional grid

cells to affect the left and right states at each of these interfaces. Recall that the

method is assumed to be stable with our choice of k and h on the regular portion of

the grid. This suggests that we should define u_ by taking the average value of U

over an interval of length h to the left of the interface z_. and define u sR by taking the

average value of U over an interval of length h to the right of zj.

For example, at z0 = 0 (the wall) we set

UoR= h (h- h')U,) (2.11) (h'Uo+

If we view the grid values as defining a piecewise constant function with values Ui

in the jth cell, then (2.11) is the average value of this function over the interval

0 _ x _ h. Note that if h' = h (the grid is completely regular) then (2.11) reduces

to u_ -- U0 as expected for Godunov's method. Recall that in Godunov's method

we take u_ = T_(U0) = 7_(u_) to impose the boundary condition v(0, t) = 0. This

suggests that more generally we take

= (ug) (2.12)

where ug is defined by (2.11). We then use the Godunov flux

= f(u (Uo,Uo)) (2.13)

5
as the flux at the wall. Using (2.12) guarantees that there will be no flux of mass or

energy through the wall and hence that the method is conservative.

To define the left and right states at zl we again construct intervals of length h

to either side of this point and average the piecewise constant function defined by U

over these intervals. To the right of zl lies a regular cell of length h and so

R

ul = UI. (2.14)

To the left of zl an interval of length h extends beyond the wall (assuming h' < h).

Beyond the wall we assume that U takes the value u_ given by (2.12). A weighted

average of this value and Uo gives u_':

1⁄4(h'u0+ (h - (2.15)

The flux ]'1 is then defined by

F1 = f(u'(u_, u_)). (2.16)

Again, if h' = h this reduces to the standard Godunov flux.

This method remains stable even when h' << h. To see why this should be so,

consider the formula (2.5) for j = 0 where hj = h'. It is the division by h' that

may cause stability problems unless the fluxes F0 and F1 themselves agree to O(h _)

as h' _ O. The Godunov fluxes based on (2.9), (2.10) do not have this property.

However, our proposed fluxes (2.13) and (2.16) do have this property, since inspection

of the formulas (2.11), (2.12), (2.14), and (2.15) shows that u_ = u_ + O(h') and

u_ = u_ + O(h') as h' --_ 0. Since the flux function f(u*(u_',ur')) is a Lipschitz

continuous function of u s and u n, it follows that F1 - Fo = O(h') as h' _ 0 and there

is at least a chance that the method remains stable for arbitrary h' << h. Numerical

experiments show that this is indeed the case (although it is possible to contrive

examples, such as a strong rarefaction wave originating at this irregularity, where the

results are not very accurate).

3 Boundary conditions in two dimensions

Turning now to the two-dimensional problem, we will give a brief description of how

the idea described above extends to handle the small cell problem.

Consider the portion of the boundary shown in Figure 2a and a typical boundary

cell (i,j). The formula for updating the value U_j is the two-dimensional analog of

(2.5),

k

U_ +1 = Uij - _[F_+I,j - F_j + Gi,j+a - Gij + Hij]. (3.17)

'_ A_
(b) Gij+l

S,_ _ (i,.i)

Fi+l_

Figure 2: (a) The Cartesian grid near the boundary. (b) Blow up of cell(ij) showing

the location of fluxes.

The fluxes F, G, and H represent flux per unit time through the corresponding side

of the grid cell (see Figure 2b) and Aq is the area of the cell. If any of the sides are

missing, the corresponding flux is zero.

On regular grid cells, Hi i = 0 and the fluxes F and G might be defined by an

extension of the Godunov method, setting

i% = hf(_,*(u___,j, u_A), G,, = hg(_,*(U_,j__, U_A). (3.18)

Here u* represents the solution to the appropriate one-dimensionM Pdemann problem

in the z or !/direction. Note that the fluxes include the factor h, the length of each

side, to give a flux per unit time across the side.

It is the denominator Aij in (3.17) that causes trouble when the cell is very small.

We again assume the method is stable on the regular portion of the grid, where

Aii= h _. To maintain stability we need to insure that our formulas for the fluxes

cause the total _ux (the sum in brackets in (3.17)) to cancelto O(_j) as _j -_ 0.

This is only possible if the fluxes axe computed via formulas that involve more than

just the two cells bordering the cell side. We take an approach analogous to what we

described above in one dimension.

Boundary fluxes. We begin by considering the boundary segment, where we

must compute the flux Hii. In two dimensions the solid wall boundary condition

requires that the normal velocity at the wall be equal to zero. If we have some value

u_ representing the value of u just inside the wM1, then we can obtain the flux Hii

by solving a one-dimensional Riemann problem in the direction normal to the wall,

with left and right states

U.L. in in
!............ •i.......................... "i.............

i : ' ...]

Figure 3: The inbox and outbox constructed from the boundary segment of cell (i, j),

and the inbox for two neighboring ceils.

The reflection operator _ is now defined by negating the normal velocity component

while leaving the tangential velocity component along with the density and pressure

unchanged. The resulting Godunov flux is used for H_j.

We obtain u_ by a procedure analogous to the one-dimensional example. We

construct a box extending a distance h away from the wall as shown in Figure 3. The

box extending into the computational domain is called inbox(i, j). The mirror image

box outside the domain is called outbox(i,j). We obtain the value u_ by viewing

the given data U as defining a piecewise constant function, constant in each grid cell,

and setting u_ to be the average value of this function over the region inbox(i,j). In

Figure 3 inbox(i,#) would contain an area-weighted average of two grid values while

the value for inbox(i + 1,j) is based on four grid values. We think of the outbox as

out in

containing the value ui_ = _(u_j).

To find the weights needed to compute z_ we must compute the intersection of the

inboxw_th each ne_byCell. This is easily accomplished with standard computational

geometry routines. Note that for a given geometry and grid these weights need only

be computed once at the beginning of the computation. They need not be recornputed

in each time step.

Fluxes at other sides. We now consider the fluxes F and G along other sides

of this cell. These are all computed by similar procedures, so to be specific we will

consider the computation of F_j, the flux on the left side of this cell.

To compute F_ we solve two Pdernann problems, one in some direction _ with

_ The some data u_, u_ and the other in the orthogonal direction 7/with data u,, u,.

choice of these directions and the data will be discussed in a moment. First we explain

how these R.iemann problem solutions are computed and used to define F_j.

Figure 4 shows a typical vertical cell interface and two orthogonal directions _ and

@

8

.._ _ . : -
L

lb_ 3

Figure 4: A vertical cell interface and the _- and r/-directions.

z/. Let 0 be the angle that _ is rotated from the z-direction (0 < 0 in this example).

Suppose we solve a one-dimensional Riemann problem in the _-direction with left and

right states u_, u_ to obtain the flux per unit length per unit time in the _-direction.

(To do this we rotate the velocity components of u_, u_ into _-7/velocity components,

solve the one-dimensional Riemann problem, and then rotate the resulting flux f back

to z-y velocity components.) Call this resulting flux f_.

Similarly, solving a one-dimensional Riemann problem in the r/-direction with

left and right states u,, u,R gives f,7, the flux per unit length per unit time in the

T/-direction. The total flux across the vertical segment of length h' is then

F = h'(h cos0 - f. sin0). (3.19)

This is the value we use for the flux Fij.

This same approach has been used by others (e.g., [4], [8], [91) to define multidimensional upwind methods. In these methods the directions _ and 7/ are chosen

based on the local flow in an attempt to use physically meaningful directions in place

of the artificial coordinate directions. For example, the direction of the velocity or

the pressure gradient might be used to define _. In our application we are only considering cells adjacent to the boundary and the relevant directions are the directions

tangential and normal to the wall. We choose _ to be the direction tangential to the

wall in one of the two cells bordering this interface. Since our primary concern is to

maintain stability in very small cells, we choose the smaller of the two adjacent cells

to define this direction. This will lead to cancellation of fluxes in tiny cells in the

same manner as previously seen in the one-dimensional example. The _/-direction is

normal to the _-direction.

Tangential boxes. We must still specify the data for these tangential and normal

R,iemann problems. We first consider the tangential problem. We use an approach

9
i

°i

\

iiiiiiiiii

.1.7-_ .._ (i, j)

<-2 ..... .............

\ .

Figure 5: (a) Tangential boxes constructed from the cellinterface.(b) Normal boxes

from the cellinterfaceand outbox(i - 1,j).

similarto the specificationof data in an £nbox described above. From the interface

we construct boxes that extend a distance h in the _-direction.Figure 5a shows an

example. The data u_, u_t is obtained by an area-weighted average of the values

in each cell the box overlaps. In our current implementation we assume the wall

isconvex, so that these boxes lieentirelywithin the computational domain. Each

box overlaps at most two grid cellsand the weights are easilycalculated. Since the

directions_ and 77 and the resultingboxes depend only on the geometry, not on the

flow variables,theseweights Can again be calculatedonce and forallas a preprocessing

step.

Normal boxes. Figure 5b shows the normal boxes in the 7/-direction.The box

in the outward directiondoes not hit the boundary and overlapsat most two regular

cells,so u,_n iscalculatedas an area-weighted avera.geofthesecellvalues.The other box

may extend beyond the boundary. Ifso, the portion lyingoutside the computational

domain llesin one or more outboxes, the artificialcellscreated in the process of

computing-t]aeboun-d-aryfl_-_ _-i/d-escribecl-above.F_Sb-shows a simple example

where the normal box intersects only one cell (i - 1,j) and outbox(i - 1,j). More

generally the normal box might intersect two ceils and their outboxes, as happens

for example when we compute the flux F_+_,j which involves cells (i,j) and (i,j - 1).

Moreover the two outboxes Wi11 in general overiap due tO the conVexity o_the region.

we again use area-weighted averaging over the four cells in question, weighting the

values Uii, Uij_I, ui°_', u_,_'l by the areas of intersection and then dividing by the

sum of all these areas.

Cancellation. Although we will not present the details here, it can be shown

that this way of defining fluxes leads to the desired cancellation of fluxes in_very small

cells. The values u_ computed at each of the three sides of a very small triangular

10

t
cell are nearly the same because of our construction. They differ by only O(Aij) as

c R and so by Lipschitz A_j _ 0. The same is true of each of the other values u_, u,7, u n

continuity of the fluxes F, G and H we obtain the required cancellation. Numerical

results show stability even when Aij is many orders of magnitude less than h 2.

Higher order methods The method (3.17) with fluxes (3.18) is only first order

accurate and is highly dissipative. In our previous work we used the wave propagation

boundary conditions together with a high resolution method away from the boundary

and obtained reasonable results (e.g., [1]). The new boundary conditions can also

be applied in conjunction with a high resolution method and gives similar results.

Moreover, with our new formulation it appears to be easier to improve the accuracy

of the boundary conditions, allowing us to obtain higher order accuracy overall. The

main idea is to introduce slopes in each cell and use piecewise linear approximations

in place of piecewise constants to define the fluxes. Near the boundary we can easily

estimate slopes in the tangential direction along the wall by differencing values in the

inboxes that we have defined above.

These improvements are still being investigated and will be reported in detail

elsewhere. Here we will only compare results obtained with the method as we have

described it and results obtained using the same interior method with the wave propagation boundary conditions described in earlier papers.

4 Numerical results

We show one representative test case, a supersonic shock going around an expansion

corner. We also show the steady state solution obtained at large times. The exact

rarefaction wave solution is a simple wave and can be computed following Section

6.17 of Whitham[11], for example.

The geometry we use is the rectangle [0, 1.32] x [0, 0.8] with a solid wall at

0.3 z < 0.1

v = 0.3(1-(x-0.1) 2) 0.1 < z < 0.7

0.192 - 0.36(x - 0.7) 0.7 <_ x <_ 1.32.

The initial conditions consist of a Mach 2.31 shock at • = 0.06 with left and right

states

and

pc = 5.1432, u c1 = 2.04511, u c=0, 2 pC=9.04545

p_ = 1.4, uR=0, 1 u R=0, 2 pR = 1.0.

11
(a)

Figure 6: Shock propagation results at t = 0.4 (a) Using the wave propagation

boundary conditions. (b) Using the new boundary conditions.

We take h = 0.02 (66 x 40 grid) and a time step k = 0.002. This corresponds to a

Courant number of roughly 0.37 relative to the regular cells with area h _. For the

crude form of Godunov's method used here, the stability restriction requires Courant

number less than 0.5. The smallest ceils near the boundary have an area roughly

10-Sh 2.

Figure 6 shows numerical results at time t = 0.4, as the shock is rounding the

corner. Results obtained with the wave propagation boundary conditions are shown

in Figure 6a, while Figure 6b shows the results obtained with our new approach.

These results are very similar. Slight discrepancies can be seen near the wall just

around the shock. For this problem both sets of boundary conditions worked weU.

We have also performed tests on other problems where the wave propagation method

shows instabilities and have observed no Such di_culties with the new method.

Figure ? shows the steady state results obtained after many iterations of the time

dependent code (no attempt has been made so far to accelerate convergence for steady

state solutions). We only show the results with our new boundary conditions. The

wave-propagation boundary conditions give very similar results. We use a coarser

grid than in the previous example (h = 0.4) in order to demonstrate that we achieve

reasonable accuracy along the boundary even with a relatively coarse piecewise 1in,

ear representation of the boundary. We also use a larger computational domain,

[0,2] x [0,1.81 to minimize the impact of the far-field boundaries. The true solution

is a rarefaction wave originating from the portion of the boundary with nonzero curvature. In the exact solution the contour lines would be straight lines. Our results

are contaminated by effects from the far-field boundary.

Near the solid wall the contour lines appear to show a boundary layer. This is an

artifact of the graphics routine, which assumes the data is on a uniform grid at cell

Z
m

10

9

8

7

6

5

4

3

0.0 0.2 0.4 0.6 0.8 1.0 1.2

(b)

1.4

Figure 7: Steady state results. (a) Pressure contours. (b) Pressure along the wall.

The solid line is the exact solution. + marks are the numerical solution.

centers. Our data near the boundary should be viewed as an approximation to the

pointwise value at the center of mass of the irregular cell, not at the center of the full

Cartesian cell.

In order to examine the accuracy at the wall, Figure 7b shows plots of the pressure

along the boundary, plotted against arclength. To obtain a boundary pressure, the

cell value U'ij and the reflected value 7_(Uij) are used to solve the one-dimensional

Riemann problem normal to the wall in each irregular cell. The resulting pressure p*

is used as the boundary pressure. Figure 7b shows these results and also the exact

solution (to machine precision) calculated using the theory of [11].

In more complicated computations we use adaptive grid refinement to obtain high

resolution results with minimal effort. The boundary conditions described here can

also be used in conjunction with the adaptive Cartesian grid code described in [1]

and

Acknowledgments. It is a pleasure to acknowledge several stimulating conversations with John Bell and Phil Colella.

13
References

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the EuIer equations in arbitrary geometries. AIAA paper AIAA'89-1930, i989.

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[3] H. H. D. CLARK_ AND M. SALAS, Euler calculations for multieliment airfoils

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[5] R. J. LEVsquB, A large time step generalization of Godunov's method.for systems of conservation laws, SIAM J. Num. Anal., 22 (1985), pp. 1051-1073.

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[lO]

[il]

i

z

=

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1. Report No.

NASA CR- 182048

[CASE Report No. 90-37

4. Title and Subtitle

2. Government Accession No.

STABLE BOUNDARY CONDITIONS FOR CARTESIAN

GRID CALCULATIONS

7. Author(s)

M. J. Berger

R. J. LeVeque

9. Performing Organization Name and Address

Institute for Computer Applications in Science

and Engineering

Mail Stop 132C, NASA Langley Research Center

Hampton, VA 23665-5225

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

3. Recipient's Catalog No.

5. Report Date

May 199'0

6. Performing Organization Code

8. Performing Organization Report No.

90-37

10. Work Unit No.

505-90-21-01

11. Contract or Grant No.

NASI-18605

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring _,gency Code

15. Supplementaw Notes

Langley Technical Monitor:

Richard W. Barnwell

Final Report

To appear in

Symposium on Computational Technology for Flight Vehicles, ed. by

Ahmed Noor, 1990

16. Abstract

We solve the inviscid Euler equations in complicated geometries using a

Cartesian grid. This requires solid wall boundary conditions in the irregular

grid cells near the boundary. Since these cells may be orders of magnitude

smaller than the regular grid cells, stability is a primary concern. We present

a new approach to this problem and illustrate its use.

17. Key Words(SuggestedbyAuthor(s))

Boundary Conditions, Euler Equations,

Cartesian Grid

18. Distribution Statement

64 - Numerical Analysis

Unclassified - Unlimited

19. SecuriW Cla_if, (of this report)

Unclassified

_. SecuriW Cla_if. (of this page)

Unclassified

21. No. of pages

16

22. Price

A03

NASA FORM 1626 OCT 86

NASA-Langley, 1990