Cranfleld
College of Aeronautics Report No. 86 26
December 1986
, . j . " - c - ' i .:...L4MEI198?
A New Numerical Technique for Quasi-Linear
Hyperbolic Systems of Conservation Laws
by
E F Toro
College of Aeronautics
Cranfield Institute of Technology
Cranfield, Bedford MK43 OAL, England
Cranfield
College of Aeronautics Report No. 86 26
December 1986
A New Numerical Technique for Quasi-Linear
Hyperbolic Systems of Conservation Laws
by
E F Toro
College of Aeronautics
Cranfield Institute of Technology
Cranfield, Bedford MK43 OAL, England
ISBN 0 947767 55 X
£7.50
"The views expressed herein are those of the authors alone and do not necessarily represent those of the Institute."
A new computational method for hyperbolic conservation laws in
one space dimension is presented. This technique is based on the
Random Choice Method (RCM). The first step is exactly as in RCM with
secondary grid, whereby intercell values at half time levels are provided.
The second step completes the solution at time level n + 1 via a finite
difference procedure in which intercell fluxes are evaluated using the RCM
solution of the first step. The method can be interpreted as a random
generalisation of Godunov's method or as a random version of the Richtmeyer
two-step Lax-Wendroff method. For the linear advection equation the
method can be shown to be second order accurate.
Application to the Euler equations, a detonation analogue (consisting
of the Burger's equation plus chemical reaction) and the linear advection
equation show that results are
\/ery
similar to those obtained by the
Lax-Wendroff method. Thus for smooth flows or even for flows containing
mild discontinuities the higher order method (HORCM) performs very
satisfactorily.
More significantly, the new method has the potential for further
development that can result in a technique capable of providing higher order
representation of smooth parts of the flow and absolutely sharp discontinuities
such as shocks and contacts. Preliminary results from applications to various
test problems are very encouraging.
1. INTRODUCTION 1
2. THE BASIC TECHNIQUE 2
2.1 Brief Review of the Random Choice Method 2
2.2 Description of the New Technique 5
2.3 Applications 6
2.3.1 Linear advection 6
2.3.2 A detonation analogue 8
2.3.3 Euler equations (Clarke's problem) 9
3. HYBRIDISATION AND SWITCHING CRITERIA 11
4. APPLICATIONS 14
4.1 Linear Advection 14
4.2 A Detonation Analogue 14
4.3 Euler Equations with Source Terms 15
4.4 Sod's Shock Tube Test Problem 15
5. CONCLUSIONS AND FUTURE DEVELOPMENTS 17
Acknowledgements 18
1. INTRODUCTION
An enormous amount of research on ways of improving computational
methods for hyperbolic problems has been taking place for some time now.
This is particularly the case of quasi-linear systems of conservation laws
of which the Euler equations are a prominent example. The existence of
solutions containing smooth parts (e.g. rarefactions) and discontinuities
(e.g. shock waves, contacts) pose, somehow, contradictory requirements on
the methods to be used for their numerical computation.
Fully second order methods (Lax-Wendroff, MacCormack) are a good
answer to smooth-flow computations, but are unsatisfactory for flows
containing discontinuities. For strongly-shocked flows these methods are
inadequate and for flows of the type encountered in combustion problems where
the structure of the discontinuity is important, alternative methods must be
used.
The Random Choice Method (RCM), formally of first order, has proved
very successful for flows involving discontinuities, no matter how strong
they are (Chorin, 1976; Gottlieb, 1986; Toro and Clarke, 1985). Great
merits of RCM are its abilities to handle all parts of the flow automatically
and provide absolutely sharp shocks, contacts and other discontinuities.
There can be randomness in the location of the discontinuities, but appropriate
sampling procedures can provide extremely accurate shock positions, say.
The limitations of RCM are more clearly manifested on the smooth parts of the
flow, in the form of randomness. For many applications the numerical noise
of RCM is tolerable, but there are cases, specially when source terms are
involved, in which the need to do something about it is clearly manifested.
In a sense, the present work can be seen as being part of that effort.
DESCRIPTION OF THE BASIC TECHNIQUE
The fundamental step of the present technique is the Random Choice Method (Chorin, 1976). We therefore briefly review this method (RCM) here.
2.1 Brief Review of the Random Choice Method
The Random Choice Method is strictly applicable to quasi-linear hyperbolic conservation laws
U^ + [""(U)]^ = S(U), with S(U) E 0
(1)
For the case of the Euler equations U and F(U) are vector functions given as follows
U
P pu E F(U) = PU pu^ + p (E + p)u(2)
The key step in the Random Choice Method is the Riemann Problem, which is defined as the initial value problem for equations (1) subject to the special data (see Fig. 1)
U(O.x)
U^, x ^ X Q
U^. X >, x^
(3)
The initial value problem (1), (3) can be solved exactly (but not in closed form) for the case of ideal gases and also in the case of gases obeying the covolume equation of state (Toro and Clarke, 1985). For more complicated equations of state approximate solutions of this Riemann problem exist (Dukowicz, 1985). RCM approximates the data at any given time by piece-wise constant functions so that a general initial value problem based on equations (1) is converted into a sequence of Riemann problems. A variety of modern numerical methods make use of the solution of the Riemann problem (e.g. Roe 1985). Pioneering work in this area is due to Godunov (1959). The Random Choice Method uses the Riemann problem solutions in a very distinctive manner.
•^.TM-
-©-
^-A
1*1
Figure 1. Data for a Riemann Problem.
Assuming we are dealing with the Euler equations then the solution
of the Riemann problem looks ias depicted by Figure 2. There are three
waves present, namely a left running wave (shock or rarefaction), a right
going wave (shock or rarefaction) and a middle wave (contact). At a given
time AT there is a range of values for the appropriate quantities (e.g. density)
that can be chosen, depending solely on the x-position. RCM picks a solution
in a 'random' fashion. The random choice process is based on a sequence of
quasi-random numbers with some desired statistical properties. So far the
most successful results have been obtained using Van der Corput sequences
(Hammersley and Handscomb, 1965).
t
rarefaction
contact surface
shock
t = t.>0
— X
There are basically two ways of implementing RCM. The most well known version makes use of a two-step procedure with second grid. This is illustrated in Figure 3(a). At the first half-time step one solves a sequenct of Riemann problems RP(i-l,i); sampling of solutions produces values to be assigned to a secondary grid denoted by the squared grid points. The secondary grid is shifted in space by Ax/2. This step gives values at time level n+1/2. The second step uses the data on the secondary
(staggered) grid to form Riemann problems the solutions of which are to be sampled in exactly the same manner as done for the previous step. Again the solution is assigned to a shifted position in space so that positions of the grid point at time level n+1 coincide with that at time level n (data level). This procedure has some disadvantages. Programming requires care in handling data correctly; also when source terms are involved the meaning of the values on the secondary grid could be objectionable, particularly when sources vary strongly with time.
Sampling
Sampling
r- Sampling
ni-V2
(a) T w o - s t e p RCM
(b) O n e - s t e p RCM
Figure 3. Two-Versions of the Random Choice Method.
An alternative approach is to proceed to sample the Riemann problem solution with data at time level n, within each computing cell as illustrated by Figure 3(b). Inthisway the solution U at time level n+1 is the result of sampling within cell i, that is the right half of the solution of the Riemann problem RP(i-l,i) and the left half of the solution of the Riemann problem
RP(i,i+l). The reader should not be left with the impression that this procedure requires more work in terms of solving Rieman problems. When proceding to update the solution at i+1 the left Riemann problem has already been solved. This was
the right Riemann problem for position i. This version of RCM is much simpler than the previous one; it is easier to code, less storage is required and for the new technique presented in this paper has some additional features as we shall see.
2.2 Description of the New Technique
The basic ingredient of the technique discussed here is a new finite difference method that uses fluxes evaluated at RCM solutions on a secondary grid that coincides with the intercell boundaries. For the purpose of
identification we shall call it HORCM, meaning higher order Random Choice Method. It is a two-step procedure. The first step is to solve all Riemann problems RP(i-l,i) and sample solutions as in the Random Choice Method with secondary grid. These solutions at time AT/2 on the intercell boundaries are denoted by u""*"? (Fiq. 3a). Now in order to advance the ies are denoted by U. f (Fig. 3 a ) .
- ,,n ._ ,,n+l .__ ..._1"!._ .,._,...„ .
solution U. to U^
one uses the f i n i t e difference scheme.
n+1 ,,n AT m / n n + i v c / i f i + J >c = u? - f i [F(u^:i) -F(u^:f)]
(4)
The procedure is illustrated in Figure 4. Notice that the solution at n+1 depends on RP(i-l,i) and RP(i,i+l). The Courant-Friedrich-Lewy condition should be observed, naturally.
Sampling for
flux evaluation
Solution obtained
by equ. (4)
n +1
n •V2
n
Figure 4. Illustration of the Higher Order Random Choice Method,
The method is effectively a random generalisation of Godunov's method (Ref. 5) which can be obtained by taking 0 = 0 in the sequence of pseudo-random numbers {0^} with 0 in [-1/2.1/2]. Godunov's method is however, a first order method, whereas HORCM can be shown, at least for
the case of the linear advection, to be second order accurate. (Toro and Roe, 1986). Computed results for a variety of problems corroborate this assertion.
HORCM can also be interpreted as a random version of the two-step Richtmeyer form of the Lax-Wendroff scheme. Here, the Lax-Friedrichs scheme
(used to obtain intermediate values for flux evaluation) is replaced by the classical Random Choice Method. The second leap-frog type step is identical to Lax-Wendroff.
2.3 Applications of HORCM
Here we apply the method to three test problems with exact solution, namely, the linear advection equation, a detonation analogue consisting of Burger's inviscid equation plus chemical reaction and the Euler equations with source terms (Clarke's problem).
2.3.1 Linear advection
The simplest hyperbolic conservation law is the linear advection equation
u^ + au^ - 0 (5) where u=u(x,t) is the conserved quantity passively convected with constant
velocity a. If the initial data
u(0,x) = UQ(X) (6)
is defined then the initial value problem (5)-(6) has exact solution
u(x,t) = UQ(X - at) (7) Example 1: half-a-sine wave.
Take the initial condition in equation (6) to be
UQ(X) = sin(Trx) 0 s< x ^< 1 (8) Figure 5 shows a comparison between the computed solution using HORCM
and the exact solution of problem (5), (8). A grid of 100 points inside the exact wave have been used (Ax = 0.01) and a = 1 in equation (5). On the whole the numerical solution is accurate, but spurious oscillations behidn the wave develop, the amplitude of which increases with time. Some
numerical diffusion is also present at the front of the rightward travelling wave. This is how the method responds to the presence of the two derivative discontinuities. We are solving the simplest example and yet the one that causes a great deal of numerical difficulties which sometimes are not present in non-linear problems. The number of time steps used or the distance
travelled by the wave relative to its width in the computations are important parameters of the test.
For comparison we solved the same problem using other well known methods under the same conditions. Fig. 6 shows the solution obtained
using the Lax-Wendroff scheme. The results are virtually identical to those of Fig. 5. The only (quite apparent) difference is that the amplitude of the spurious oscillations produced by the present method are smaller than those of the Lax-Wendroff technique. Fig. 7 shows the results of the
Godunov's method, which is a special case of the HORCM presented here. Severe 'clipping' and smearing of the discontinuities are well known features of
Godunov's method. Fig. 8 shows the results obtained by the classical Random Choice Method. Admitedly, these are surprisingly good. One tends to expect randomness in the numerical respresentation of smooth solutions. Falle (1985) used the same test problem for a third order multigrid technique under same computational conditions but with solution convected for 4000 time steps. Fig. 9 shows HORCM solution for that number of time steps. Degradation of solution as time increases is more visible now. Fig. 10 shows the
corresponding Lax-Wendroff result. The solutions of Figs. 9 and 10 are virtually identical, but itisnow more obvious that the amplitude of the
oscillations produced by HORCM is smaller than that of Lax-Wendroff. This will also be the case inthenext test problem.
Example 2: squared wave,
Consider the initial condition for equation (5) to be
' 0, X ^ 0
u,(x) 1, 0 < x><; 1 (9)
Figure 11 shows the results using HORCM after 1000 time steps and for computational parameters described on the Figure. Overshoots and oscillations are then expected features from a higher order method on problems as (5), (8). Figure 12 shows the corresponding Lax-Wendroff result. Again results are very similar; spurious oscillations are slightly less severe in the HORCM result.
2.3.2 A detonation analogue
Here we consider a detonation analogue consisting of the inviscid Burger's equation with chemical reaction, namely the 2 x 2 system
Pt + Px = 0 (10)
A^ = r(A) (11)
with 'equation of state'
P = HP' + AQ) (12)
The steady case has exact solution. This can be very useful in
testing numerical methods. More details about this test problem can be found in Clarke et. al. (1986). The problem (10), (11) contains some of the fundamenal features of the physical problem of detonations in condensed media. One of them is the presenceof a strong shock leading the detonation front. Another is the presenceof a reaction zone attached to the leading shock. These two features are closely interrelated and they easily defeat most current
numerical techniques used in their simulation. The correct structure of the reaction zone (a very thin region) depends crucially on the ability of the numerical technique to resolve the shock wave adequately. The Random Choice Method provides absolutely sharp shocks but looses accuracy within the
reaction zone (smooth). In fact the randomness is enhanced by the presence of the source term governing reaction.
HORCM can represent the reaction zone accurately even on a coarse grid, but as indicated before the response to the shock wave is inadequate. Figure 13 shows computed results for the detonation analogue with the steady exact solution as the initial condition. Here p represents the 'density' and X is the reaction progress variable. The pressure p is then calculated via the 'equation of state'. The first full line is the initial profile. Computed
profiles at later times are shown by symbols with the exact solution
superimposed for comparison. A grid of ten points for the reaction zone has been used. As already mentioned HORCM responds to the shock with 'random' overshoots. This is precisely the aspect of HORCM we want to modify (see sections 3 and 4 for hybrydised version).
2.3.3 Euler equations with source terms
Here we apply HORCM to the unsteady Euler equations in one space dimension. We consider a test problem involving source terms. The formulation, solution and meaning of the problem can be found elsewhere
(Clarke and Toro, 1985 and in Clarke 1986).
The problem is given by equations (l)-(2) in which the inhomogeneous term S(U) is p/a up/a S(U) & + Ju^)P/a (13)
where C is a constant, a is the sound speed and y is the specific heat ratio.
The test problem resembles the physical action of an igniter venting gases inside a tube. Assuming uniform atmospheric conditions at time zero and that the 'igniter' acts only on the left half of a 1 m long tube, the problem can be solved exactly. The initial conditions as given are not essential for obtaining the analytical solution. The resulting test problem can be ^^^ry useful for testing numerical methods/programs. Computed
results are shown in Figures 14-16 for y = 1-4, C = 10 for times (in miliseconds) as indicated. As expected the solution is very accurate in smooth parts of
the flow, unlike the one obtained by the traditional Random Choice Method. In this test problem there are three regions likely to provoke difficulties to almost any method. The first zone is that around the head of the expansion travelling leftwards; there is a discontinuity in derivative there which will be rounded by a given amount depending on the method. One can say that HORCM performs quite satisfactorily there. The second region is located around the interface where a discontinuity of the source terms is present. The solution exhibits very large gradients which are more apparent for the velocity (see Figure 15). HORCM responds with oscillations there; overshoots in velocity and undershoots in density and pressure can be seen. This is not too serious
but it would still be desirable to improve on this (see sections 3 and 4 of this report). The third region is around the rightward propagating front; again there is a discontinuity in derivative in the solution there. Some smearing is present in the HORCM solutions of Figs. 14-16.
For comparison we also show the corresponding RCM solutions under the same computational conditions as before. The results are shown in Figures 17-19. As usual RCM is quite inaccurate in the smooth (non-uniform) parts of the flow and it is precisely this feature of the traditional RCM we are trying to
improve, while preserving its unique shock capturing capability (see sections 3 and 4 of this report). Figures 20-22 show the results obtained by the Godunov's method, which, as stated earlier is a limiting case of the present HORCM. On the whole the solution is good in the smooth parts of the flow but very inacccurate near discontinuities in derivative, i.e. the first and third regions referred to above.
All three methods applied to Clarke's problem in this section have different and complementary features; a suitable combination (hybridisation) of these methods can lead to a rather good technqiue for hyperbolic systems. Also, it should be realised that all of these three techniques are very closely related.
The Higher Order Random Choice Method presented here has been shown to be accurate for smooth flows and also for some discontinuities in derivative (regions one and three for Clarke's problem). For very high gradients such as in region two (x = 0 in Figures 14-22) oscillations occur. For the case of weak shocks HORCM does rather well. This was tested using Clarke's problem for later times when a shock forms on the right hand side. For strong shocks (or other discontinuities) however, HORCM has problems that are similar to those of the Lax-Wendroff scheme as illustrated by the application in section 2.3. In the next section we propose some hybridisation procedures designed to improve the performance of HORCM for all discontinuities.
3. HYBRIDISATION AND SWITCHING CRITERIA
The Higher Order Random Choice Method (HORCM) presented here can be
used in practical applications with results that are similar to those of the
Lax-Wendroff scheme. For smooth flows the method is accurate, but near shocks
and other discontinuities its performance is unsatisfactory. In this section
we present some preliminary ideas designed to recover the 'sharp shock'
capability of RCM and some of the features of the low order Godunov's method.
First let us indulge ourselves on rather speculative observations that
are to be the subject of further research. The present method as it stands
can be seen as the upper limit of a range of algorithms that begin with
Godunov's method as the lower limit. The upper limit (HORCM) is order 2 while
the lower limit (Godunov) is order 1. Intermediate schemes result from
contracting the interval of 'random' sampling I^ = [-1/2. 1/2]. If 0n
is a member of a Van der Corput sequence then T0^, with T in [0,1], is a
a member of a modified sequence in a contracted interval of sampling I^.
The conjecture that follows is that the order k in [1,2] varies in some
fashion with T. Godounov's method is given by T=0 and the present HORCM
method is given by T=l. It would be of interest to prove this and find out
the shape of k = k(T). Most certainly, the local Courant number also plays
a role in the order function k. For identification purposes let us call these
schemes T-schemes.
A first statement we can make is that for mild discontinuities such as
discontinuities in derivative one should drop the order of HORCM to an
intermediate scheme in the family of T-schemes. Computed results have
indicated to us that dropping straight down to Godunov (T = 0) is a drastic
measure. Also, dropping the order by using RCM around mild discontinuities
is not the best solution. Using a member of the family of intermediate T-schemes
appears to be a good idea. In fact one can think of an adaptive-order method
within the family of T-schemes bounded by Godunov (T = 0) and the HORCM (T = 1 ) .
This is however an area to be explored by careful analysis and tests before a
fully practical method can be achieved.
For strong discontinuities (e.g. shocks and contacts) there is no
doubt, one must revert to traditional RCM. This is a very easy and natural
procedure to implement in practice. Numerical results support this argument.
Another important aspect of the hybrid method is the choice of a
switching criterion that is sufficiently general, reliable and relatively
cheap to implement in practical applications. Here we present an approach
based on a non-dimensionalised length.
A non-dimensional length as indicator of flow changes
Consider the quantity
'i Ax ^3x'9x^^
^^ '
where Ax is the mesh size in space,u is any of the unknowns of the problem
(e.g. conserved variables) and i refers to position in space and n to time
level.
Using central difference approximation to derivatives in (13) we have
u" , - u"
l"? . J [ 1±1 lzl_] (15)
1 •- n o n n -^
^ '
"i+1 - 2^- + '^i-i
Now define
n n
^1 n n
^ '
^• - ^-1
this quantity is a popular parameter used in various methods to detect the
nature of flow changes. Using q. in equation (15) we have
n n q? + 1
l" = l" (q-) = J [ 4 ] (16)
' ' ' q- - 1
Equation (16) says that for neighbouring states
^^_-,, u^,
u^* , that are
comparable (i.e. q^ » 1 ) , l"? are an indication of flow changes. Regions
of slow change can be treated well by using HORCM. Small values of l"? are
to be taken as indicative of rapid change; this means that either RCM or
some T-scheme, w i t h T e [0,1], should be used.
Noting that l'j'(O) = -i, l" > J as q" ^ », and l" ^ i as q" -> -» a
small value of the length means something close to J, This automatically
defines a lower limit for variation of q., namely, zero. As to the upper
limit for q"? we do not yet know the answer, but values between 2 and 3 appear
to be sensible. In fact a flux-limiter type of analysis (Ref. U ) suggests
that
I/YIJ",where y? "is the local Courant Number, can be taken as an upper
limit,
Once rapid changes have been detected it remains to scrutinise these more closely to discriminate between mild and strong discontinuities. The explicit solution of the Riemann problem will then complete the information needed.
Some applications of the hybrid method and shown in the next section.
4. APPLICATIONS
Here we present computed results applying some of the preliminary
ideas discussed previously. The results show that the HORCM method
presented in this report can be suitably extended to deal effectively
with discontinuities, no matter how strong they are.
4.1 Linear Advection
Here we apply the hybrid method to the two examples considered in
section 2.3.1 namely the sine wave and the squared wave.
Fig. 23 shows the computed solution for the sine wave after 4000 time
steps. Compare now with Fig. 9 which shows the result obtained by HORCM
above. See Fig. 10 too for comparison. The hybrid-method solution has no
oscillations; there is no clipping either, although some smearing is still
present.
Fig. 24 shows the computed result for the squared wave (plus the exact
solution given by broken lines). Compare to Figs. 11 and 12. The
solution is very good indeed.
4.2 A Detonation Analogue
Here we return to the problem discussed on section 2.3.2.
Fig. 25 shows computed results for ten different times. The first
profile is the initial profile given by the exact steady solution. For
subsequent times both the exact and computed solutions are plotted for
comparison. Note that only 10 points within the reaction zone have been used.
The shocks are absoutely sharp and the representation of the smooth parts
of the flow is
very
accurate. Compare to results of Fig. 13 obtained by
HORCM alone. Overshoots and smearing near shocks have completed disappeared.
An important point to make here is the accuracy of the shock positions.
Careful observation of Fig. 25 tells us that the error in shock position is
not greater than a cell length. Other computed results confirm this
empirical observation.
Fig. 26 shows an amplified plot of the initial profile (exact steady
solution) for this problem, together with the length function (absolute value)
given by equation (15). Large values of 1.° are set to 10. Note how the
two discontinuities in the solution are indicated by very small values
(close to i) of the length function. The uniform parts of the length function
should actually have the value infinity (l!^(l)).
4.3 Euler Equations with Source Terms
Here we use the hybrid method to compute solutions to Clarke's problem dealt with in section 2.3.3. Figs. 27-29 show results for density, velocity and pressure for 3 different times. Compare with Figs. 14-16, which are the results obtained by HORCM alone. Notice the improvement of solution near X = 0. Oscillations have disappeared, although the pick values in velocity
(Figs. 15 and 28) are not exactly attained. For the results of Figs. 27-29 Godunov's Method (T = 0) has been used to deal with the high gradients near X = 0. Here is where the use of an T-scheme option (with, T £ [0,1])
discussed before can prove to be a good answer.
For comparison see results of Figs. 17-19 obtained by the Random
Choice Method alone. Also, see results of Figs. 20-22 where Godunov's method alone has been used.
Next we apply the hybrid method to a problem containing strong discontinuities.
4.4 Sod's Shock-Tube Test Problem
This problem has become a popular test problem for algorithms designed to solve the Euler equations in situations involving strong shock waves and contacts. Here the initial data is p, = 1.0, p = 0.125, u, = 0, u = 0, p, = 1.0, p = 0.1 together with y = 1.4 (specific heat ratio).
Figs. 30-32 show results for density, velocity, pressure and internal energy for 3 different times. This is the standard case run on a 1 m long tube with initial discontinuity at x = 0.5.
For comparison, the reader is referred to Sod (1978) where several numerical techniques are tested on this shock-tube test problem. Also it is useful to compare the present results with those obtained by the MUSCL code of Van Leer (1979). The present method has a clear advantage in dealing with shocks and contacts. Elsewhere in the flow, our results compare well with those of Van Leer's. Our slight oscillations in density behind the contact (more visible in the internal-energy plot) are the result of curing undershoots produced by HORCM near the tail of the rarefaction by using RCM. Again we feel that the use of an T-scheme should be the correct answer. We also run the case of a 2 m long tube with the initial discontinuity at x = 0.8. Computed results
For comparison we solved the same problem using the Random Choice Method
alone. Results are displayed in Figs. 37-40. The quality of the RCM results
is not surprising. For later times however the randomness within the rarefaction
is quite appreciable, this randomness is even worse when source terms are
involved as illustrated by applications to Clarke's problem (see section 2.3.3
of this report).
5. CONCLUSIONS AND FUTURE DEVELOPMENTS
A new Higher Order Random Choice Method for quasi-linear hyperbolic
systems has been presented. The method performs well for the case of
predominantly smooth flows and can be used in practical computations.
Also we show some advances towards a hybridised version of HORCM
with encouraging results, although the present state of the resulting
algorithm is not yet a tool for practical applications.
The hybrid method has basically three closely interrelated components:
HORCM, Godunov's method and RCM. Use of traditional RCM for strong
discontinuities (shock waves and contacts) is the correct answer to the problems
experienced by HORCM for this type of flow. For mild discontinuities such as
discontinuities in derivative one can use Godunov's method, but we are not
entirely convinced about this. Use of RCM for these types of discontinuities
is not advisable.
A nondimensional-length approach for detecting discontinuities and
other rapid changes in the flow has been presented. This can be used to
decide as to which option of the hybrid method is to be used. Results
are very encouraging but the associated switching criterion still requires some
further refinements.
The hybridised version of HORCM has been shown to have a lot of
potential and we feel that some more research on this can lead to a very good
numerical technique for solving problems of technological interest.
We also speculate about the existence of a family of schemes of the
type of HORCM presented here. For identification purposes we called its
members T-schemes (with T a parameter in [0,1]). The family has clearly
defined bounds, namely, Godunov's method (T=0) and the HORCM presented in this
report (T=l). The order k = k(T) should vary in some fashion between 1 and 2.
Use of these T-schemes for mild discontinuities (discontinuities in
derivative) appears to be a good answer to the problems experienced by HORCM,
Computed results confirm this assertion. This is still a rather obscure
area in which further research is needed.
Acknowledgements
The author gratefully acknowledges the useful ideas emerging from
fruitful discussions within the CFD group in the College of Aeronautics.
In particular thanks are due to Professors J F Clarke and P L Roe.
References
Chorin, A. 1976
Random Choice Solutions of Hyperbolic Systems. Journal of
Computational Physics. 22, 517-536, 1976.
Gottlieb, J. 1986
Lecture Course notes on Random-Choice Method for solving one-dimensional
unsteady flows in ducts, shock tubes, and blast-wave simulators.
AC-Laboratorium Spiez, Switzerland, 21-30 May 1986.
Toro E.F. and Clarke J.F. 1985
Application of the Random Choice Method to computing problems of
solid-propellant combustion in a closed vessel.
CoA Report NFP85/16 College of Aeronautics, Cranfield Institute of
Technology, UK.
Dukowicz J.K. 1985
A general, non-iterative Riemann solver for Godunov's method.
Journal of Computational Physics, 61, 119-137, 1985
Godunov, S.K. 1959
Mat. Sb. 47, 271, 1959. Also as USJPRS translation 7226 (1960).
Hammersley J.M. and Handscomb D.C. 1965
Monte Carlo Methods, Methuen, London, 1965
Roe P.L. 1985
Some contributions to the modelling of discontinuous flows.
Lectures on Applied Mathematics, Vol. 22, 163-193, 1985
Clarke J.F., Roe P.L., Simmonds L.G., and Toro E.F. 1986
Numerical studies of a detonation analogue. CoA Report NFP86/26,
November 1986, College of Aeronautics, Cranfield Institute of
Technology, UK.
Clarke J.F. and Toro E.F. 1985
Gas flows generated by solid-propellant burning. Proc. Symp. Numer.
Simulation of combustions Phenomena. Lecture Notes in Physics, 241,
192-205 Glowinski, Larrouturou and Teman (Edts.), Springer-Verlag.
10. Clarke J.F. 1986
Compressible flow produced by distributed sources of mass: an exact solution.
To appear in Journal of Fluid Mechanics. 11. Toro E.F. and Roe P.L. 1986
A hybridised higher-order random choice method. To appear. 12. Falle S.A.E.G. 1985
Numerical Gas Dynamics.
Cosmical Gas Dynamics pp 149-162, F.D. Kahn (Editor), 1985, VNU Science Press.
13. Sod G.A. 1978
A survey of several finite difference methods for systems of non-linear hyperbolic conservation laws.
Journal of Computational Physics, 27, 1-31, 1978. 14. Van Leer B. 1979,
Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's Method.
7. 1—4 O _! •> 5 3. t 15 22 —r" 25 30 35 "40 ?5" DISTANCE 50 55 60 G5 X10" — I 70
THE LINEAR ADVECTION EQUATION
FIGURE 8 ;EXACT (FULL-LINE^ AND COMPUTED (SYMBOL> SOLUTIONS; RANDOM CHOICE METHOD USED
DX=0.01, CFLNO--3.50, A --i.00, ^INAL T T M E STEP-^ 2000
5® CTI 00 >. ID IT •«- rf) (N ,UlJO"liA (S S) u (S -CN • I N I » C9 CD a LÜ CO -D rr. u a' l i j a o I (Y Lb O S) ^ , (S X «^ •z :^ >' o n ^ i _ I - I-<- ^ c'> o o Lü Lb CT ~ 2" LL •> O Q' <-Lb 3r Lb X t-o m
r
•2-(j-^ '^ LÜ S u •D -a '' r o <-/ ^ • Lb 2" C9 IT' Ó <• l' Lu Q Lb a~ v55 '
i
*^fa n l ^ ^ T 9 ^ • f F ^ X •-.^ n J > ^ P ^ \ %3„ % : -^ V i nV
= r ^ sr^"N
^ ' ^s<°
^ - - .s ^
' ^ - ^ l ö " ^'>>-0 ^ ï 6 ^ 1 • " c T - - ^ ^ r_ 4
^ / -j-^ '•--.^,^ ^ . . ^s
'^ 1 I 1 \1
3
, i j JT^ ' T rk
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P P co er 00 f^ (0 if) < Cj fN - s — -<s V 'M N (55 -N Lu U •z •f u 0 ' a i Cl 53 iN (N IS tN (S CN or V — — ' . u i J C i a A C l Lb Q O i— Lu r u Cia
z
LL. < - CS t 1 ^ " '^ ,, o o CL . ..-. Lu L. l. L <- - 3 ''" ~ j _ . ! o o Lu Lb LI- TT '" 2 - - *-O _ ' •-• O - - , t- ai X L. r Z' Lb • - , - • • > CO u C l >. «r a s cy Lu CS Lu D -2' a. •' - i r _' o ' ' c_. L b - f f-"" - ^ 1 - ^ CS < - LI"-' CS U '' Zi 2 ' - • ü \ Ll ' 'w CS u L. CS < i' Lu Q O 1—1 Lb C f —1 CZJ u5), '-n <?3 00 t o - ^ a * (S 'M I A J . I Ü 0 * I 3 A •D
a
o
X I -Lu u u o cy oz
Lb 1 CS CS .. (Sz
-n
Z - • '' o •- a ' _ ' ' -<" c:) CJ' r) co o Lu Lu ' " ^ Z O ^ o ai -•' 27 •-- ",— L u V-• >a o
< ' Lb Z O' <r Lu 3 a r o C_/ CS CSa
Lu 2 -X < Lui (S ir CS i'o
z
z l i l J ^ o O' m 1— CJ <-V Lb ( LL C J » • ~~ ( S • CS ll V a Lb (ir o10 12 DISTANCE
18
DISTANCE
DETONATION ANALOGUE, PROBLEM 1
FIGURE 13 :. COMPUTED SOLUTIONS (SYMBOL) AND EXACT SOLUTION GIVEN BY FULL-LINE PROFILES. TF= 2.1-^9, Q= 2 5 . 0 0 , CFL NUMBER=1 ,09
X10I 22^ 2 0 . 18. 16. 14. (/I \ r >: 12 u o lil > 10. 8. 6. 4. TIME = TIME --TIME = 0 . 1 0 0 MS 0 . 2 5 0 MS 0 . 4 0 0 MS - 2 5 H — l É f a — • a g i ^ i ' i S ' i ' i y i f t -20 - 1 5 - 1 0 - 5 i^<^>iiM^ l ^ T M M i W +(*! 1111 1111, X A X I S 10 15 20 25 X10''^
FIGURE 15: CLARKE'S TEST PROBLEM.
X10~' 50-, TIME -•^5 •t0 35. r UJ [2 30. 25. 20. 10 I I I 1 1 1 I I 1 1 1 1 1 I I I H+. m*iViNi*WWViitWWV<4WiVilVfW I 5 >M^t**MMMJM>l^k*MM^MMMMMM^MMM V TIME + TIME 0 . 100 MS 0 . 2 5 0 MS 0 . 4 0 0 ns i • 1 1 1 T laaae - 2 5 - 2 0 - 1 5 - 1 0 - 5 ^ 5 X AXIS
FIGURE 16: CLARKE'S TEST PROBLEM.
X10~l 4 0 . 38 3G 34 32 30. to è 2 8 ^ 26 lU Q AWW^iMiWWWVVVVI^IilWW 24. 22. 20. 18 IG. 14 12. I I I I I I I I I I I I I I M I I I I ^ A A A ^ ^ M • H * l * * > * l > i * l fcA^»AAXfc»,MhfcfcAAAAi»fcAfcl A T= 0 . 1 0 0 MS V T= 0 . 2 5 0 MS + T= 0 . 4 0 0 MS - I r - 2 5 - 2 0 20 X10~'2 25
FIGURE 17: CLARKE'S TEST PROBLEM.
X10' 20 18. 16 14. ^ 12J u g 10 UJ > 8. 6. 4. 2. -25 -20 TIME = 0.100 MS TIME = 0.250 MS TIME = 0.400 MS X AXIS 15 20 25 X10''2
FIGURE 18: CLARKE'S TEST PROBLEM.
X10~' 50., A TIME = 0.100 MS V TIME -- 0.250 MS + TIME = 0.400 MS • W W W W W I I I I I I I I I
FIGURE 19: CLARKE'S TEST PROBLEM.
X10"' 40, 38. 36 34 32. 30-r 28. ^ 2G UJ o 24 mVii^^H*f**NiV^iiVtW 22. 20 18. 16 14 1111111111 fr»^— U,tMMMMMMMMMMMMM*M*MMM^ A T= 0.100 MS V T= 0.250 MS + T= 0.400 MS -25 -20 -15 -10 X AXIS
FIGURE 2 0 : CLARKE'S TEST PROBLEM.
XI 0 20 IS 16. >-12, 10 8= 2= A TIME = 8.100 MS 7 TIME -- 0.250 MS + TIME 5^ 0.400 MS
=2i
PIGURf
tl i
CLARKI'S TIST PROBLin.
X10~l 5 0 , 45. 40. 35 LU o: in ^ 30 UJ 25. 20. ^^^^+Hu ^ W W W W ^ W V ^ W W W W W i 1 5 . >>*•>**Mdyyüyfc*..i>*>ihAA<^^AA<i»>*.ii***^ A TIME = 0 . 1 0 0 MS y TIME = 0 . 2 5 0 MS r i r i E = 0 . 4 0 0 MS T 1 \ 1 1 Wa^^i^M^ ^ " S p ( W , M M M j i * ' H t | i l 11 i|i I I , 2 5 - 2 0 - 1 5 - 1 0 - 5 ^ 5 10 15 20 25 X AXIS )(.\<d~^
FIGURE 22: CLARKE'S TEST PROBLEM.
u o Ui 8 R 4 ? 1^ 2 j5 / J"
J
§
d d d d d °l ^ D / 0 / ff' ^ ' J F / ff ' / \ ff
s
1
} T 0 a ^ q a óA
°\
a\ o\ n\ • V DV D\ o\ D\ °n\°d
a
la T L ^ ^ ^ ^ ^ ^ ' ^ * " h 195 200 205 210 DISTANCE 215 220 X10 -1THE LINEAR ADVECTION EQUATION
FIGURE 23 lEXACT (BROKEN-LINE) AND COMPUTED (SYMBOL) SOLUTIONS; HYBRID METHOD USED
DX=0.01, CFLNO=0.50, A -1.00, FINAL TIME STEP= 4000
8
t 6
o o _ } Ui > 4 2 0 1 i .1
1
1 50 - 55 G0 - 65 DISTANCE ^ " ^ ~ % 70THE LINEAR ADVECTION EQUATION
FIGURE 24 :EXACT (BROKEN-LINE'» AND COMPUTED (SYMBOL) SOLUTIONS; HYBRID METHOD USED, D X = 0 . 0 1 , CPLNO-0.50, A - 1 . 0 0 , "^INAL TrriE STEP-- 1000
m c= Z 70 m CD -< c\> T l (jy C • • | - X — -< a 1 CD m i— ; o - 1 —. H-, o z o z m > :z - 1 "D m 1—1 73 - 1 o O X Z Tl O —• O > n z m LO > tn o 1— • (~ o C CT - 1 - 1 C T l --< m il o -— z K) co T l ;o -^ ^ o CD co IX) N ^ t -- 3 m UJ zi o o il r" —' ^^ K) cn > • z S) o (S " m X o > T l O n -H z co e o 3 n 00 c m - i 73 1— Il O O z <J1 (S (S — N) , w ® ( UI IvJ-CD ( U l C0< w-D ' •—• CO > -- i U l > z o ( m cn-Ul (7)i U I 0 ' , co -*• Ul c n ï i CS r^^^ ïw — ^ " " ' ^ • " ^ s ^ — ^ " " • ^ « ^ ^ ^ 1 .v^ L P * s t ^ ^ ^ * * * S i i 1 ^''**"**^i^ ^ * » ^ ^ ^ ^ i ^ ^ ^ ^ 4 ^ cn ^ ^ * ^ ' « S i . ^ ^ ^
' ' ^ ' - ^
K-^ K-^ " K-^ K-^ K-^ S i ^ ^ * * < s . - ^ = * CJ-( l ^ * ^ * * ^ S i ^ ^ * - * i f i i (*) P**<a j ^^'^ös^ **«. '"' ^ * * > t t ^ —l U l ^ ^ » « A ^ > ^ ^ ^ 1 ^ i a ^ m uiJ ^ ^ * ^ < j . ^ ^ * ^ i m i . , Ul ^*«-^ ^*«-^ « S i ^*«-^ ^^*s»„^ ^
?-^- - 2 » '^'V i ^
^ ^ * s „ - ^ L—«— ^ — r-1 — ^ ^ * * • 'V G. co z LU Q 15 20 25 "30" 35 — 1 — 40 45 50" DISTANCE 55 G0 65 « — X 1 0 - ' 70 10* z o z li-o z lU _1 8. 6. 0 . . —p-5 10 15 20 25 30 35 40 45 50 DISTANCE 55 60 65 X10~' 70
DETONATION ANALOGUE, PROBLEM 1
FIGURE 2 6 : I N I T I A L PROFILE FOR DENSITY AND CORRESPONDING LENGTH FUNCTION. Q= 2 5 . 0 0
WITH GODUNOV*S OPTION FOR HIGH GRADIENTS
X10" 4 0 , 1111111111111111 n+y ^ T= 0 . 1 0 0 MS V T= 0 . 2 5 0 MS + T= 0 . 4 0 0 MSFIGURE 27: CLARKE'S TEST PROBLEM.
WITH GODUNOV^S OPTION FOR HIGH GRADIENTS
X10' 20 16 14. <C 1 2 , > - I-I—) O ° 10 UJ > 0. TIME = 0 . 1 0 0 MS TIME -- 0 . 2 5 0 MS TIME = 0 . 4 0 0 MS -25 -20 - 1 5FIGURE 28: CLARKE'S TEST PROBLEM.
WITH GODUNOV^S OPTION FOR HIGH GRADIENTS
X10~' 5 0 , \JM>IMMM*MMMMMMMMM*M*MMM*^ - 2 5 ^^20 H 5 H 0 ^ TIME = 0 . 1 0 0 MS TIME = 0 . 2 5 0 MS TIME = 0 . 4 0 0 MS X AXIS X10"2FIGURE 29: CLARKE'S TEST PROBLEM.
10., 9-8 . 7.
t 6.
Q 4. 3. 2. 1. 0 i ) 1 X 1 0 " l 10. 9. 8. 7. PRESSUR E U l < n 4. 3. 2 . 1. 0 .e
iiiiiiiiii 1 1 FIGUREp
l
l
1
te 1 b t 2 3 4 5 6 7 8 9 10 X AXIS X 1 0 - ' oTIME STEP : 35 } ) )L
2 3 4 5 6 7 8 9 10 X AXIS X 1 0 - ' 10 9. 8. 7, >-t G. u o üJ 5. > 4 . 3. 2. 1. 0 0 ) • } 3 (1 \. 0 1 2 3 4 5 6 7 8 9 X AXIS X 1 0 " ' oTIME STEP : X10~1 'ia 28 26. S 24, UJ z UJ 22. 20 18. 16.e
\ \ \L
I 1 1 2 3 - ^ 5 6 7 8 9 X AXIS X 1 0 - 'SOD'S SHOCK-TUBE TEST PROBLEM (STANDARD CASE) 30: COMPUTED SOLUTION BY THE HYBRID METHOD
(SYMBOL) AND EXACT SOLUTION (FULL L I N E )
V
10
35
CD z UJ Q T 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 X AXIS X 1 0 " ' CJ o _J L U > lU (/) to UJ a.
oTIME STEP : 70 oTIME STEP 70
0 1 2 3 4 5 6 7 8 9 10 X AXIS X10-'
0 1 2 3 4 5 6 7 8 9 10 X AXIS X10"'
SOD'S SHOCK-TUBE TEST PROBLEM (STANDARD CASE)
FIGURE 31: COMPUTED SOLUTION BY THE HYBRID METHOD
10, 9. 8 . 7. g 5_ Q 4 . 3 J 2. 1 . 0. E ) XI 0 10 9. 8. 7. UJ ? G. in to lU er o: 5 4. 3 2. 1. 0.
e
- 1 1 FIGURE ~ \ \ \ ^UJikiiinQ) 2 3 4 5 6 7 8 9 10 X AXIS X 1 0 " ' oTIME STEP : 105 « ^V
\ \ \ \ niiimiTn) 2 3 4 5 6 7 8 9 10 X AXIS X 1 0 - ' 10. 9. 8. 7 t G. u o UJ 5 . > 4. 3. 2. 1. 0, ii
/I
/ƒ
/ƒ
) 1 2 3 4 5 6 7 8 - 9 10 X AXIS X10-1 oTIME STEP .• 105 X10 30 28. 26. S 24. Of UJ 22. 20. 18. 16. 2 - 1 ^ \ \ iimu 1 2 3 4 5 6 7 8 9 10 X AXIS X10-1SOD'S SHOCK-TUBE TEST PROBLEM (STANDARD CASE) 32: COMPUTED SOLUTION BY THE HYBRID METHOD
(SYMBOL) A ND E. <ACT SOI _UTION (1 =ULL L I N E )
> - I-cn z UJ • 10 12 14 16 18 20 X AXIS X 1 0 " ' 1 0 , H. G. o o _ J LU > 3. 2. 8 10 12 14 IG 18 20 X AXIS X 1 0 - ' LU CD CD LU CU CL X 1 0 ~ l 10.amn
o TIME STEP : 35 TIME STEP : 35
9, 8 7, 6 5 4 . 3 2. nmmmiB XI 0 30. 2 8 26 - 1 """""""""""™" ^ 24 Q; UJ z m 2 2 . 20. 18. {QmBusfl 0 ~2 4 S 8 10 12 14 16 18 20 X AXIS x i 0 - > 16 -I 1 1 1 r 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10-'
SOD'S SHOCK-TUBE TEST PROBLEM
FIGURE 33: COMPUTED SOLUTION BY THE HYBRID METHOD
10 9 . 8. 7. DENSIT Y cn c n 4 . 3 . 2 . 1. 0. i ^ i 2 4 X10 10., 9. 8. 7. PRESSUR E cn c n 4. 3. 2. 1 0. 2 -1 2 4 FIGURE 34:
1
\ \ \ '%naim ) e 8 10 12 14 16 18 20 X AXIS X10""' oTIME STEP : 70 ? \ \ \V
6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X 1 0 - ' 9 l 8 . 7 . y-o o _1 e; LU ^ -> 4 . 3. 2 . 1.. 0.f
f
i
1
1^
f
f
/ V 0 2 4 6 8 1 0 1 2 1 4 1 G I 3 20 X AXIS X10--' V o TIME STEP : 70 X 1 0 - ' 30 28. 26. S 24. Of LU Z lU 22. 20. 18. 16..e
ARMBaanu y ) 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS 0 9 - 1SOD'S SHOCK-TUBE TEST PROBLEM COMPUTED SOLUTION BY THE HYBRID METHOD (SYMBOL) AND EXACT SOLUTION (FULL L I N E )
to z LU Q 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X 1 0 - ' o o UJ > 0 2 4 6 8 10 12 14 1G 18 20 X AXIS x i 0 - > l U =5 CD C/1 UJ or
o TIME STEP 105 TIME S^EP ; 105
0 2 4 6 8 1 0 1 2 1 4 1 G 1 8 20 X AXIS X10-1
0 2 4 6 8 10 12 14 iG i.9 20 X AXIS xi0-'
SOD'S SHOCK-TUBE TEST PROBLEM
FIGURE 35: COMPUTED SOLUTION BY THE HYBRID METHOD
_ ± 0 , 9_ 8 . 7. ^ 6 . DENS I U l ' t , 3. 2. 1. 0. E \ \ \ ^ 1 1 i 2 4 X10 - ^ 1 9. 8. 7 LU
? 6.
C/5 CT' UJ cr Of 5 . Q. 4. 3. 2. 1 0.e
-1 \ ?j ^ \ 1 1 2 4 FIGURE 3 6 : ^\
msoDU) 1 1 1 1 1 1 1 1 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10-1 oTIME STEP : 140l
\
immoj 6 8 1 0 1 2 1 4 1 G 1 8 20 X AXIS X 1 0 - ' 10^ 9. 8. 7. t G. o > 4. 3-2. 1 PIi
/ / / /ƒ
/1
1
J
0 2 4 6 8 1 0 1 2 1 4 1 6 18 20 X AXIS X i 0 - 1 o TIME STEP : 140 X10 30 28. 26. G 24. UJ z UJ 22. 2 0 . 18. 16.e
-1\
\
[ 1 ' 1 1 1 ' ' t 1 1 2 4 6 8 1 0 1 2 1 4 1 G 1 3 20 X AXIS X 1 0 - 'SOD'S SHOCK-TUBE TEST PROBLEM COMPUTED SOLUTION BY THE HYBRID METHOD (SYMBOL) AND EXACT SOLUTION (FULL L I N E ^
> I -CD Z LU Q 10^ 9 , 8. 7. 6. 5. 4. 3. 2. 1. 0 £ ) 2 4 6 >
I
8 X 1 ba 10 12 A) <IS 14 16 18 20 X 1 0 - ' >I -1-4u
o
_1 UJ > 10, 9 8. 7. 6 5. 4. 3. 2 0. 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10 •1 X10~' 10«iBBno TIME STEP : 35 o TIME STEP 35
9. 7. LU Of (D CD UJ cr a: j J 3. 2. 1. > -(0 LU Z UJ I I I I I 1 I I I 1 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10-1 8 10 12 14 16 18 20 X AXIS X10-1
SOD'S SHOCK-TUBE TEST PROBLEM
FIGURE 37 : COMPUTED SOLUTION BY RCM ON FIXED GRID
CO
z
UJ Q >-U O LU > 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10""1 B H P H W H P 8 10 12 14 16 18 20 X AXIS X10~1 UJ Oi. w CD LU Oi.O TIME STEP 70 o TIME STEP : 70
>-cs Of LU z UJ X AXIS X I 0 " 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS x t 0 - '
SOD'S SHOCK-TUBE TEST PROBLEM FIGURE 38: COMPUTED SOLUTION BY RCM ON FIXED GRID
to UJ a 8 10 12 14 16 18 20 X AXIS X10-1 u o Ul > 10 12 14 16 18 20 X AXIS X10-1 UJ w w UJ Of CL
O TIME STEP : 105 o TIME STEP : 105
>-(0 Q; m z UJ 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10~1 0 2 6 8 10 12 14 16 18 20 X AXIS X10-'
SOD'S SHOCK-TUBE TEST PROBLEM
FIGURE 39: COMPUTED SOLUTION BY RCM ON FIXED GRID
10. 9. 8. 7. DENSIT Y -• U l C O 3-2 . 1 . 0 { ) 2 4 X10 9. 8. 7. UJ % G. (D CD LU cr 4. 3 2. 1. 0. E -1 2 4 FIGURE 4 0 : \ ( ( Qoomo) 6 8 10 12 14 16 18 20 X AXIS X10-' oTIME STEP : 140
V
. BIIBPfli 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X 1 0 - ' 10 9. 8. 7. >-t G. u o _J i : UJ 3 -> 4. 3. 2. 1. 1^ / ^ / /ƒ
/ /_ƒ
0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10-1 oTIME STEP : 140 X10 30 28. 26. S 24. UJ z lU 22. 20. 18. 16.e
-1 \ \ \ \ 1 BQQQJ 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 X AXIS X10-'SOD'S SHOCK-TUBE TEST PROBLEM COMPUTED SOLUTION BY RCM ON FIXED GRID (SYMBOL) AND EXACT SOLUTION (FULL L I N E )
