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The random projection method for sti multi-species detonation capturing Weizhu Bao and Shi Jin y School of Mathematics, Georgia Institute of Technology Atlanta, Georgia, GA 30332, USA Abstract

In this paper we extend the random projection method, proposed by the authors for general hyperbolic systems with sti reaction terms [1], for underresolved numerical simulation of sti, inviscid, multi-species detonation waves. The key idea in this method is to randomize the ignition temperatures in suitable domains. Several numerical experiments, in both one and two dimensions, demonstrate the reliability and robustness of this novel method.

Nomenclature Bk e L M N p qm T Tk u v wm Wm zm k

a constant in the frequency factor for the k-th reaction total energy the number of dierent kinds of atoms in all species the number of reactions the number of species pressure the heat of formation for the m-th species temperature ignition temperature for the k-th reaction x-component velocity y-component velocity m-th species source term molecular weight of the m-th species mass fraction of the m-th species exponent of the temperature dependence of the frequency factor for the k-th reaction cp to cv ratio

Email address: [email protected]. On leave from: Department of Applied Mathematics, Tsinghua University, Beijing 100084, P. R. China y Email address: [email protected]. Address after July 1, 2000: Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706; email: [email protected]. Research supported in part by NSF grant No. DMS-9704957 

1

mn 0 mk 00 mk 

the number of the n-th atom in a molecule of the m-th species stoichiometric coecient for the m-th species appearing as a reactant in the k-th reaction stoichiometric coecient for the m-th species appearing as a product in the k-th reaction density

1 Introduction Consider the reactive Euler equations that model the time-dependent ow of inviscid, compressible, multi-species reacting ows [11], [21]

Ut + F (U )x + G(U )y = S (U ); where

0 BB BB BB BB U =B BB BB @

0 v 1 0 u 1 BB uv CC BB u + p CC BB v + p CC BB uv CC CC B CC BB CC ; F (U ) = BB (e + p)u CC ; G(U ) = BBB (e + p)v BB vz CC BB uz CC BB vz CC BB uz CC C B@


C B


A @


A vzN uzN zN  u v e z z

2

2

1

1

1

2

2

2

with

wm = Wm

M X k=1

( 00

mk

? 0

mk ) Bk

T k e?Tk =T

and

N X m=1

N z !jk Y j ; W j j 0

=1

zm = 1 :

(1.1)

1 0 CC BB CC BB CC B CC ; S (U ) = BBB CC BB CC BB CA B@

1 CC CC CC CC ; (1.2) CC CC


CA wN 0 0 0 0 w w

1 2

1mN;

(1.3) (1.4)

The pressure for ideal gas is given by     (1.5) p = ( ? 1) e ? 21  u + v ? q z ? q z ?


? qN zN ; the temperature is de ned as T = p=. Equations (1.1)-(1.4) are referred to as the multi-species reactive Euler equations with Arrhenius kinetics. We will also consider (1.1)-(1.3) with 2

2

1

1

Bk T k e?Tk =T replaced by the Heaviside kinetics

Bk T k H (T ? Tk ); where H (x) = 1 for x > 0 and H (x) = 0 for x < 0. 2

2

2

One of the main numerical challenges for chemically reacting ows is that the kinetics equations (1.1) often include reactions with widely varying time scales. The chemical time scales, as characterized by Bk? , may be orders of magnitude faster than the uid dynamical time scale. This leads to severe problems of numerical stiness. Even a stable numerical scheme may lead to spurious unphysical solutions unless the small chemical scales are fully resolved numerically. Developing robust numerical methods for sti reacting ows has been a very active area of research in the past two decades. In particular, many works have contributed to the analysis and development of underresolved numerical methods which are capable of capturing the physically relevant solutions without resolving the small reaction scales. Of course, when one does not resolve the chemical scale numerically (using grid size larger than the width of the reaction zone), it is impossible to capture the detailed structure (such as the pressure spike) of the reaction zone. Thus the best one can hope is to capture the speed of discontinuity as well as other features of uid dynamics. In the two-species case, i.e. the reaction of type A ! B , it was rst observed by Colella, Majda and Roytburd [8] that an underresolved numerical method leads to a spurious weak detonation wave that travels with an incorrect speed. It is known that the smeared numerical shock pro le, which exists for all shock capturing methods, leads to a too early chemical reaction once the smeared value of the temperature in the numerical shock layer is above the ignition temperature [8], [17]. Since then, lots of attention have been paid to study this peculiar numerical phenomenon (see [4], [17], [19], [13], [5]) for the two-species case, and several remedies are available in the literature, for examples [3], [9], [15]. Recently we proposed the random projection method as a general and systematic method to solve hyperbolic systems with sti reaction term, applicable to reacting ow problems with two-species [1]. The origin of a random method can be traced back to the classical work of Glimm [12], and later Chorin successfully adopted it {called the random choice method{ for reacting ow computation [7]. However, this classical method is of a Godunov type method relying on the solution of the Riemann problem. When applying it to the reacting ows one needs to obtain the solution of the generalized Riemann problem for hyperbolic systems with source terms [4]. The random projection method we proposed is a fractional step method that combines a standard { no Riemann or generalized Riemann solver is needed { shock capturing method for the homogeneous convection with a strikingly simple random projection step for the reaction term. In the random projection step, the ignition temperature is chosen to be a uniformly distributed random variable between two stable equilibria. When the random sequence is chosen to be the equidistributed van der Corput's sampling sequence [14], we proved, for a model scalar problem, a rst order accuracy on the shock speed if a monotonicity-preserving method {which includes all TVD schemes{ is used in the convection step [1]. A large amount of numerical experiments for one and two dimensional detonation waves demonstrate the robustness of this novel approach for two-species reactions [1],[2]. In engineering applications, chemically reacting ows often involve more than two species. Sometimes there are tens to hundreds of species [23], [22]. Usually implicit or semi-implicit methods were used in application when one also considers the viscosity terms, see [6], [10], [18] and references therein for details about resolved calculations. For inviscid ows, underresolved numerical calculation also leads to spurious nonphysical 1

3

wave, as shown by one of the numerical examples in Section 5. In this paper, we extend the random projection method to sti multi-species reacting

ows in both one and two space dimensions. Extensive numerical experiments will be conducted to examine the eectiveness and robustness of this novel method. The paper is organized as follows. In Section 2 we review brie y the random projection method, proposed by the authors [1], for a scalar model problem. In Section 3 we introduce the random project method for the multi-species problem (1.1) in one space dimension. In Section 4 this method is extended to two space dimension. In Section 5 several numerical examples, in both one and two space dimensions, will be presented. We end in Section 6 with some concluding remarks.

2 Review of the random projection method for a scalar model problem In this section we review the random projection method, introduced in [1], for the following hyperbolic conservation law with sti reaction term, ut + f (u)x = ? 1" (u ? )(u ? 1); ?1 <  < 1; (2.1) with piecewise constant initial data 2

(

u(x; 0) = u (x) = 1?;1; 0

xx ; x>x :

(2.2)

0

0

Here " is the reaction time, f is a convex function of u, i.e. f 00 (u) > 0, and x is a given point. The source term in (2.1) admits three local equilibria, i.e., the unstable one u = , and the stable ones u = 1. When the solution is at equilibria, the reaction term has no eect. Thus the exact solution is a shock discontinuity connecting u = 1 with u = ?1 and propagating to the right with the speed determined by the Rankine-Hugoniot jump condition [17]: s = 21 [f (1) ? f (?1)] : (2.3) Namely, ( 1; if x  x + st; u(x; t) = ? (2.4) 1; if x > x + st: 0

0

0

Let h be the spatial increment and k be the time step, such that x is a grid point, i.e. x = l(0)h with l(0) an integer. The numerical solution is evaluated at the points (jh; nk), j = 0; 1; 2;


, n = 0; 1; 2;


. Let unj approximate u(jh; nk) and un be the solution vector of u(
; nk) at time t = tn = nk. When the reaction term is resolved, i.e. k = O(h) n; if uj  n;

S (k) : unj = 1?;1; +1

for all j:

(2.6)

In this method, one random value of n will be selected per time step. The combination of the two steps gives the numerical scheme

Sgp(k) :

un = S (k)Sc(k)un: +1

(2.7)

The stability condition for this method is the usual CFL condition determined by the convection term. Here and in our practical computations, we always use van der Corput's sampling sequence for fn g. The merit of this sequence is that it produces an equidistributed sequence on the interval [0; 1], and among all known uniformly distributed sequences its P m deviation is minimal [14]. Let 1  n = k ik 2k , ik = 0; 1, be the binary expansion of the integer n. Then the van der Corput's sequence is de ned on [0; 1] as: =0

#n =

m X k=0

i k 2? k ; ( +1)

n = 1; 2;


:

(2.8)

We rescale it in order to get a random number generator n on [?1; 1]:

n = 2#n ? 1;

n = 1; 2;


:

(2.9)

The sequence fn; n = 1; 2;


g is equidistributed on the interval [?1; 1]. We have proved [1] that the random projection method (2.7), when a monotonicitypreserving method Sc { which includes all TVD methods { is used for the convection term, can capture the correct location of discontinuities for the scalar model problem (2.1)-(2.2) with a rst order accuracy. Let l(n) ? l(0) be the number of grid points that the discontinuity has traveled after n time steps.

Theorem 2.1 [1] Given T > 0. Let Sc be a monotonicity preserving method. The

dierence between the shock location of the exact solution, x0 + stn , and the numerical one, l(n)h, as determined by the random projection method (2.7), has the following estimate:

jx + stn ? l(n)hj  C (T )hj ln hj; 0

(2.10)

for any 0 < tn  T , and xed  = sk=h, where C (T ) is a positive constant depending on j ln(T )j.

5

This shows that the shock location is captured with a rst order accuracy for all time, although the reaction time " is not resolved numerically. In [1], [2] this method was generated to two-species sti detonation wave computation, where the key idea is to randomize the ignition temperature in a suitable domain. In the following two sections, we will extend the random projection method to one and two dimensional multi-species detonation computation.

3 One space dimension In this section, we shall describe the random projection for the sti multi-species   n method T n n n reacting ows in one space dimension. We use Uj = j ; (u)j ; ej ; (z )nj; (z )nj;


; (zN )nj to denote the approximate solution of U = (; u; e; z ; z ;


; zN )T at the point (xj ; tn). Our main interest is an underresolved numerical method which allows k = O(h) >> (1=Bk ), yet still obtains physically relevant numerical solutions. Consider the Riemann initial data 1

1

2

((x; 0); u(x; 0); p(x; 0); z (x; 0); z (x; 0);


; zN (x; 0)) ( = ((l ((xx));; uul((xx));;ppl ((xx)); ;((zz))l ; (; z(z)l);
;



;
(;z(Nz)l)) ;) ; r r r r r N r 1

2

2

1

2

1

2

if x  x ; if x > x ; 0

0

(3.1)

where x is a given point and PNm (zm )l = PNm (zm )r = 1. In addition the initial mass fraction should satisfy the following condition: 0

=1

=1

N  (z ) N  (z ) X X mn m l mn m r = Wm Wm ; m m =1

=1

1  n  L:

(3.2)

The above equalities guarantee the conservation of the number of atoms in the reactions. We also assume that after the reaction, at least one species disappears (fully reacted). Thus there exists an j such that (zj )l = 0 for each reaction. This implies that for all m = 1; 2;


; N:

(wm)l = 0;

(3.3)

This assumption clearly holds outside the reaction zone. Since an underresolved method ignores the reaction zone of width O( Bk ), each species has only two physical states, either (zm )l or (zm )r . Without loss of generality the data in (3.1) are chosen such that the discontinuity, initially at x = x , moves to the right. The case when the discontinuity moves to the left can be treated similarly. The random projection method is a fractional step method 1

0

S (k) : 1

U n = Sp(k)SF (k)U n +1

(3.4)

that consists of solving the homogeneous convection

Ut + F (U )x = 0; 6

(3.5)

by a standard shock capturing method, denoted by SF (k), for one time step, followed by a random projection step for the sti chemical reaction terms t = 0; (u)t = 0; et = 0; (z )t = w ; (z )t = w ;


; (zN )t = wN : (3.6) Due to (3.3), the above ODE collapses to a simple projection ( if Tj  mink Tk ; for all j ; n (zm )j = ((zzm ))l ;; (3.7) if Tj > maxk Tk ; m r with U  = SF (k)U n , Tj = pj =j being the values after the convection step. However, on the discrete spatial domain, due to the grid eect, this projection (referred to as the deterministic projection) yields an incorrect speed, since the smeared value of the temperature T  in the detonation layer, once above maxk Tk , will trigger the chemical reaction too early. The key idea in the random projection method is to use a randomized ignition temperature in a suitable domain. This strategy could be successful since the speed of the front does not depend on the speci c value of Tk , as far as it is in the range between the equilibrium states Tl and Tr on both sides of the detonation. We now describe the details of this random projection step. Let pr (x; 0) ; n = (Tl ? Tr )#n + Tr ; Tl = xx0 r (x; 0) l (x; 0) where #n is the van der Corput's sequence, see (2.8) fro detail. Thus n is a equidistributed sequence over the domain [Tl ; Tr ]. Assume x = l(0)h is a grid point. Since the projection always make zm either (zm )l or (zm )r for all 1  m  N , therefore, at any time step tn , there is an l(n) = j , j an integer, such that ( if j  l(n); 1  m  N : n (zm)j = ((zzm ))l ;; (3.9) if j > l(n); m r Here l(n) is the location of the jump for zmn . The random projection will be performed around x = l(n)h, a procedure referred to as the local random projection in [1]. Let d be the estimated number of smeared points in the detonation layer. Modern shock capturing methods always introduce few smeared points across the discontinuity. The information on d can easily obtained from numerical experience (normally between 1 and 5). We now move the jump of zm according to the following algorithm: Sp(k) : n =  ; mn = m ; en = e; set l(n + 1) := l(n) ? 1; For l = l(n) ? 1;


; l(n) + d; do  l(n + 1) ( := l; if Tl > n ; if j  l(n + 1); (zm )nj = ((zzm ))l ;; for all m; j: (3.10) if j > l(n + 1); m r In the above algorithm, only d + 2 points will be scanned. The stability condition for the algorithm (3.4) is the usual CFL condition determined from the convection step SF (k). 1

1

2

2

+1

0

0

0

+1

+1

+1

+1

7

For numerical comparison, we also use the deterministic method which discretizes the chemical reaction term (3.6), with Tk xed as given by the original problem, by an implicit method (for example, backward Euler method) and solving the corresponding nonlinear system using Newton's method.

4 Two space dimension In this section, the random projection method is extended to two-dimensional multispecies reacting ows (1.1). For simplicity, we consider the ow in a two-dimensional channel. Let the initial data be ((x;(y; 0); u(x; y; 0); v(x; y; 0); p(x; y; 0); z (x; y; 0); z (x; y; 0);


; zN (x; y; 0))  (y); = ((l ;; uul;;vvl ; ;ppl ; (; z(z)l); (;z(z)l);
;



;
(;z(Nz)l )); ); ifif xx  (4.1) >  (y); r r r r r r N r 1

1

2

2

1

2

where  (y) is a given function of y. These data are chosen such that the discontinuity moves to the right and with the same assumptions on (zm )l and (zm)r as in the 1-d case. Let n = (Tl ? Tr )#n + Tr ; Tl = pl ; Tr = pr ; (4.2) l r and #n being the van der Corput's sampling sequence on the interval [0; 1] de ned in (2.8). Let the grid points (xi; yj ) = (ih; jh); i; j =


; ?1; 0; 1;


, with equal mesh spacing h. The time also uniformly  spaced with time step k.  nlevel tnn = nkn, kn = 0; 1;n2;


are n n n Let Uij = ij ; (u)ij ; (v)ij ; eij ; (z )ij ; (z )ij ;


; (zN )ij be the approximate solution of U = (; u; v; e; z ; z ;


; zN ) at (xi; yj ; tn). Similar to the one-dimensional case, the random projection method is a fractional step method that consists of a standard shock capturing method for the homogeneous convection 1

1

2

2

Ut + F (U )x + G(U )y = 0; (4.3) denoted by SFG(k) for one step, followed by a random projection step for the sti chemical reaction terms t = 0; (u)t = 0; (v)t = 0; et = 0; (z )t = w ; (z )t = w ;


; (zN )t = wN ; (4.4) where Tk , the ignition temperatures, are randomized in suitable domains. Notice that, at any time step, for each j , there is an lj (n) = jn, jn an integer, such that ( if i  lj (n); 1  m  N : n (zm )ij = ((zzm ))l ;; (4.5) if i > lj (n): m r Here lj (n) is the location of the jump for the numerical solution of zm along the line y = yj at time tn. Let U  = SFG(k)U n, and d be the estimated number of smeared points in the detonation layer. Then the random project algorithm for the chemical reaction term (4.4) follows: S p(k) : n =  ; un = u; vn = v; en = e ; 1

2

+1

1

+1

2

+1

8

2

+1

For j do Set lj (n + 1) := lj (n) ? 1; For r = lj (n) ? 1; lj (n);


; lj (n) + d; do lj (n + 1) := r; if Trj > n; ( if i  lj (n + 1); (zm )nij = ((zzm ))l ;; if i > lj (n + 1); m r

for all m; i: (4.6)

+1

The combination of the two steps gives

U n = S p(k) SFG(k) U n:

S (k) :

+1

2

(4.7)

2

The stability condition for this algorithm is still the usual CFL condition determined from the convection step SFG(k).

5 Numerical examples In order to verify the performance of the random projection method for sti multi-species detonation computations, we conduct several numerical experiments in both one and two dimensions. In our computation, the operator SF (k) and SFG(k) are the second order relaxed scheme [16], which is a TVD scheme without the usage of Riemann solvers or local characteristic decompositions. We choose d = 5 in (3.4), (4.7) and use the Heaviside kinetics in (1.3) in our computations in this section. It is known that the stiness problem is more severe in the Heaviside case than the Arrhenius case [9]. In these tests, the hydrodynamic data (such as pressure, mass, velocity) are choosen arti cially rather than from physical experiments.

Example 5.1: We consider a reacting model 2 A + B ! 2 C: A prototype reaction for this model is 2 H + O ! 2 H O: 2

2

2

In this problem, there are three species and one reaction. The parameters are: M = 1, N = 3, = 1:4, T = 2:0, B = 10 ,  = 0, q = 100, q = 0:0, q = 0:0, W = 2, W = 32, W = 18,  0 ; = 2,  0 ; = 1,  0 ; = 0,  00; = 0,  00; = 0,  00; = 2. The initial data are piecewise constants given by 1

2

3

6

1

11

1

21

1

31

2

11

(

3

21

31

if x  2:5; if x > 2:5;

(; u; p; z ; z ; z )(x; 0) = ((l ;; uul;;ppl; ;((zz))l ; (; z(z)l); (;z(z)l)); ); r r r r r r 1

2

1

3

2

1

3

2

1

3

where pl = 20:0, l = 2:0, ul = 8:0, (z )l = 0:0, (z )l = 0:0, (z )l = 1:0; and pr = 1:0, r = 1:0, ur = 0, (z )r = , (z )r = , (z )r = 0:0. This problem is solved on the interval [0; 50]. 1

1 9

2

8 9

1

2

3

9

3

The exact solution consists of a detonation wave, followed by a contact discontinuity and a shock, all moving to the right. We obtain the \exact" solution by an explicit method (the second-order relaxed scheme for the convection terms followed by forward Euler method for the chemical reaction terms) using a ne mesh size h = 0:0025 (i.e. 20001 grid points on the interval [0; 50]) and a small time step k = 0:0001. This is a resolved calculation. We compare the results obtained by the local random projection method (3.4) and the deterministic method using a coarse mesh h = 0:25 (i.e. 201 grid points on the interval [0; 50]) and large time step k = 0:01, and output the numerical solutions at t = 4:0. Figure 5.1(a) shows the numerical solution by the random projection method (3.4), while Figure 5.1(b) shows the numerical solution by the deterministic method. It can be seen that the random projection method captures the correct speed of the discontinuity even when the reaction scale is not numerically resolved. The shock and the detonation front are captured with a higher resolution than the contact discontinuity, a typical phenomenon for shock capturing methods. The location of zm may be few grid points away from the exact location due to random eect, but such a deviation does not grow in time. There are small post shock statistical uctuations due to the random nature of the method, but they are at an acceptable level. The deterministic method produces spurious nonphysical waves, as was observed in earlier literatures for sti reacting ows with two-species. In all of the following examples, the deterministic method always produces spurious nonphysical waves when the reaction scale is not resolved. We will not report those results, and will only present the solutions obtained by the random projection method.

Example 5.2: The set up of this example is similar to that in Example 5.1 except that here q = 1000, B = 500,  = 1, ul = 10:0, pl = 40:0, (z )l = 0:325, (z )l = 0:0, (z )l = 0:675, (z )r = 0:4, (z )r = 0:6, (z )r = 0:0. The exact solution consists of a detonation wave, followed by a contact discontinuity and a shock, all moving to the right. We obtain the \exact" solution similarly as that in Example 5.1. Figure 5.2 shows the numerical solutions by the random projection method (3.4) with h = 0:25 (i.e. 201 grid points on the interval [0; 50]) and k = 0:01 at time t = 3:0. All discontinuities are captured numerically. 1

1

3

1

1

1

2

2

3

Example 5.3: We consider a reacting model A + 2 B ! C + 2 D: A prototype reaction for this model is

CH + 2 O ! CO + 2 H O: 4

2

2

2

In this problem, there are four species and one reaction. The parameters are: M = 1, N = 4, = 1:4, T = 2:0, B = 10 ,  = 0, q = 500, q = 0:0, q = 0:0, q = 0:0, W = 16, W = 32, W = 44, W = 18,  0 ; = 1,  0 ; = 2,  0 ; = 0,  0 ; = 0,  00; = 0, 1

1

2

6

1

3

4

1

1

11

10

2

21

3

31

4

41

11

 00; = 0,  00; = 1,  00; = 2. The initial data are ( (; u; p; z ; z ; z ; z )(x; 0) = ((l ;; uul;;ppl ; ;((zz))l ; (; z(z)l); (;z(z)l); (;z(z)l )); ); r r r r r r r 21

31

41

1

2

3

1

4

2

1

3

2

if x  2:5; if x > 2:5;

4

3

4

where pl = 40:0, l = 2:0, ul = 10:0, (z )l = 0:0, (z )l = 0:2, (z )l = 0:475, (z )l = 0:325; and pr = 1:0, r = 1:0, ur = 0, (z )r = 0:1, (z )r = 0:6, (z )r = 0:2, (z )r = 0:1. The exact solution consists of a detonation wave, followed by a contact discontinuity and a shock, all moving to the right. We obtain the \exact" solution similarly as that in Example 5.1. Figure 5.3 shows the numerical solutions by the random projection method (3.4) with h = 0:25 (i.e. 201 grid points on the interval [0; 50]) and k = 0:01 at time t = 3:0. All discontinuities are captured numerically. 1

2

1

3

2

4

3

4

Example 5.4: We consider a reacting model A + B ! 2 C; 2 C + A ! 2 D; and the species E appears as a catalyst. A prototype reaction for this model is

H + O ! 2 OH; 2

2 OH + H ! 2 H O;

2

2

2

and N appears as a catalyst. In this problem, there are ve species and two reactions. The parameters are: M = 2, N = 5, = 1:4, T = 2:0, T = 10:0, B = 10 , B = 2  10 ,  =  = 0, q = 0:0, q = 0:0, q = ?20:0, q = ?100:0, q = 0:0, W = 2, W = 32, W = 17, W = 18, W = 28,  0 ; = 1,  0 ; = 1,  0 ; = 0,  0 ; = 0,  0 ; = 0,  0 ; = 1,  0 ; = 0,  0 ; = 2,  0 ; = 0,  0 ; = 0,  00; = 0,  00; = 0,  00; = 2,  00; = 0,  00; = 0,  00; = 0,  00; = 0,  00; = 0,  00; = 2,  00; = 0. The initial data are 2

1

1

2

1

3

2

4

3

5

22

32

42

52

32

42

52

5

21

11

31

21

(

5

1

4

11

22

2

41

31

41

4

2

1

2

51

12

51

12

 2:5; (; u; p; z ; z ; z ; z ; z )(x; 0) = ((l ;; uul;;ppl ; ;((zz))l ; (; z(z)l); (;z(z)l); (;z(z)l ;)(z; (z)l )); ); ifif xx > 2:5; r r r r r r r r 1

2

3

4

1

5

2

1

3

2

4

3

5

4

5

where pl = 40:0, l = 2:0, ul = 10:0, (z )l = 0:08, (z )l = 0:72, (z )l = 0:0, (z )l = 0:0, (z )l = 0:2; and pr = 1:0, r = 1:0, ur = 0:0, (z )r = 0:0, (z )r = 0:0, (z )r = 0:17, (z )r = 0:63, (z )r = 0:2. The exact solution consists of a detonation wave, followed by a rarefaction wave and a shock, all moving to the right. We obtain the \exact" solution similarly as that in Example 5.1. Figure 5.4 shows the numerical solutions by the random projection method (3.4) with h = 0:25 (i.e. 201 grid points on the interval [0; 50]) and k = 0:01 at time t = 3:0. All waves are captured numerically with the correct speeds. 1

2

5

4

1

3

2

4

3

5

The above examples demonstrate that the random project method works eectively for one dimensional sti multi-species detonations, although the reaction scales are not numerically resolved. It captures the correct speeds of detonations, as well as other standard uid dynamical structures with high resolutions. The deterministic method, however, always produces spurious waves when the chemical scale is not numerically resolved. 11

Example 5.5: This is a two-dimensional example with radial symmetry analogous to Example 5.3, i.e. the parameters M , N , , T , B ,  , q ,


,  00; are the same as those 1

1

1

1

41

in Example 5.3 except q = 100. A similar example, but with only two-species, was used in [15]. The initial values consist of totally burnt gas inside of a circle with radius 10 and totally unburnt gas everywhere outside of the circle. Furthermore the unburnt and burnt states are chosen in a way analogous to the one dimensional case, i.e. 1

(; u; v;( p; z ; z ; z ; z )(x; y; 0) = ((l ;; uul (;x;v y; )p; v; l((zx;)y;)(; zpl;)(;z(z)l ;)(z; ()zl ;)(z);)l ; (z )l ); r r r r r r r r 1

2

3

4

1

p

1

2

2

3

3

if r  10; if r > 10;

4

4

where r = x + y , pl = 40:0, l = 2:0, ul = 10x=r, vl = 10y=r, (z )l = 0:0, (z )l = 0:2, (z )l = 0:475, (z )l = 0:325; and pr = 1:0, r = 1:0, ur = 0:0, vr = 0:0, (z )r = 0:1, (z )r = 0:6, (z )r = 0:2, (z )r = 0:1. This is a radially symmetric problem and the important feature is that the detonation front is circular. This problem is solved on the domain [0; 50]  [0; 50] with h = 0:5 and k = 0:01. Solid wall boundary conditions are used along x = 0 and y = 0. Out ow boundary conditions are used along x = 50 and y = 50. Figure 5.5 (a) shows the velocity elds and Figure 5.5 (b) shows pro les of the pressure p, temperature T , and 100 times the mass fraction of the rst species, 100z , (here we show 100z not z itself, for better visualization) on the line y = x (x  0:0) by the random projection method (4.7) at time t = 1, t = 2, t = 4 and t = 6 respectively. It can be seen that the detonation front remains circular and no spurious nonphysical wave is generated when using the random projection method (4.7). On the other hand, if one uses the deterministic method, the detonation front does not remain circular and spurious nonphysical wave is generated if the same grid size and time step are used. 2

2

3

2

1

2

4

1

3

4

1

1

1

Example 5.6: This is a two-dimensional example analogous to Example 5.4, i.e. the parameters M , N , , T , T , B , B ,  ,  , q ,


,  00; are the same as those in Example 1

2

1

5.4. The initial data are given by

2

1

2

1

52

(; u; v; p; z ; z ; z ; z ; z )(x; y; 0) ( = ((l ;; uul;;vvl ; ;ppl ; (; z(z)l); (;z(z)l); (;z(z)l ;)(z; (z)l ;)(z; ()zl )); ); r r r r r r r r r 1

2

3

4

5

1

2

1

where

3

2

4

3

if x   (y); if x >  (y);

5

4

5

(

:5 ? jy ? 12:5j; jy ? 12:5j  7:5;  (y) = 12 5; jy ? 12:5j > 7:5; and pl = 40:0, l = 2:0, ul = 10:0, vl = 0:0, (z )l = 0:0, (z )l = 0:0, (z )l = 0:17, (z )l = 0:63, (z )l = 0:2; and pr = 1:0, r = 1:0, ur = 0:0, vr = 0:0, (z )r = 0:08, (z )r = 0:72, (z )r = 0:0, (z )r = 0:0, (z )r = 0:2. This problem is solved on the domain [0; 150]  [0; 25] with h = 0:5 and k = 0:01. Solid wall boundary conditions are used along y = 0 and y = 25. In ow boundary conditions are used along x = 0 and out ow boundary conditions are used along x = 150. One important feature of this solution is that the triple points along the detonation front 1

4

2

5

3

2

3

1

4

5

12

travel in the transverse direction and bounce back and forth against the upper and lower walls, forming a cellular pattern. Behind the detonation front, there is a very strong shock. Figure 5.6(a) shows density contours calculated by the random projection method (4.7) at several dierent times. Figure 5.6(b) shows pro les of pressure p, temperature T , and 300 times the mass fraction of the rst species, 300z on the line y = 12:5 at times t = 2, t = 4, t = 6 and t = 8. One can see that no spurious nonphysical wave is generated when using the random projection method (4.7). However, if one uses the deterministic method, spurious nonphysical wave is generated if the same grid size and time step are used. 1

6 Conclusions In this paper we extend the random projection method to underresolved computation of sti multi-species detonations. This method is based on the random projection method proposed by the authors for general hyperbolic systems with sti reaction terms [1]. The key idea of this method is to randomize the ignition temperatures in a suitable domain. Numerical experiments in both one and two dimensional problems demonstrate that this method, although very simple and ecient, provides physically correct solutions with a high resolution when the small chemical scale is not numerically resolved.

References [1] W. Bao and S. Jin, The random projection method for hyperbolic conservation laws with sti reaction terms, J. Comp. Phys., submitted. [2] W. Bao and S. Jin, The random projection method for sti detonation waves, SIAM J. Sci. Comp., submitted. [3] A. C. Berkenbosch, E. F. Kaasschieter and R. Klein, Detonation capturing for sti combustion chemistry, Combust. Theory Modeling 2, 313-348 (1998). [4] M. Ben-Artzi, The generalized Riemann problem for reactive ows, J. Comp. Phys. 81, 70-101 (1989). [5] A. Bourlioux, A. Majda and C. Roytburd, Theoretical and numerical structure for unstable one-dimensional detonations, SIAM J. Appl. Math. 51, 303-343 (1991). [6] T. R.A. Bussing and E. M. Murman, Finite-volume method for the calculation of compressible chemically reacting ows, AIAA J. 26, 1070-1078 (1987). [7] A. J. Chorin, Random choice methods with applications to reacting gas ow, J. Comp. Phys. 25, 253-272 (1977). [8] P. Colella, A. Majda and V. Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Stat. Comp. 7, 1059-1080 (1986). 13

[9] B. Engquist and B. Sjogreen, Robust dierence approximations of sti inviscid detonation waves, UCLA CAM Report 91-03 [10] P. Gerlinger, P. Stoll and D. Bruggemann, An implicit multigrid method for the simulation of chemically reacting ows, J. Comp. Phys. 146, 322-345 (1998). [11] I. Glassman, Combustion, Academic Press (1987). [12] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18, 697-715 (1965). [13] D. F. Griths, A. M. Stuart and H. C. Yee, Numerical wave propagation in an advection equation with a nonlinear source term, SIAM J. Num. Anal. 29, 12441260 (1992). [14] J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, Methuen, London, 1965. [15] C. Helzel, R.J. LeVeque and G. Warnecke, A modi ed fractional step method for the accurate approximation of detonation waves, SIAM J. Sci. Comp., submitted. [16] S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48, 235-276 (1995). [17] R. J. LeVeque and H. C. Yee, A study of numerical methods for hyperbolic conservation laws with sti source terms, J. Comp. Phys. 86, 187-210 (1990). [18] H. N. Najm, P. S. Wycko and O. M. Knio, A semi-implicit numerical scheme for reacting ow, J. Comp. Phys. 143, 381-402 (1998). [19] R. B. Pember, Numerical methods for hyperbolic conservation laws with sti relaxation. I. spurious solutions, SIAM J. Appl. Math. 53, 1293-1330 (1993). [20] M. A. Sussman, Source term evaluation for combustion modeling, AIAA paper 930239. [21] F. A. Williams, Combustion Theory, Addison-Wesley, Reading, MA, 1985. [22] G. J. Wilson and M. A. Sussman, Computation of unsteady shock-induced combustion using logarithmic species conservation equations, AIAA J. 31, 294-301 (1993). [23] H. C. Yee and J. L. Shinn, Semi-implicit and fully implicit shock-capturing methods for nonequilibrium ows, AIAA J. 27, 299-307 (1989).

14

pressure

density

50

4

45 3.5

40 35

3

30 2.5

25 20

2

15 1.5

10 5

1

0 −5

0

10

20

30

40

0.5

50

0

temperature

10

20

30

40

mass fractions of z and z 1

50

2

14 0.9 12

0.8 0.7

10

0.6 8

0.5 0.4

6

0.3 4

0.2 0.1

2

0 0

0

10

20

30

40

−0.1 30

50

35

40

45

50

Figure 5.1: Numerical solutions of Example 5.1 at t = 4:0 calculated with h = 0:25, k = 0:01. { : `exact' solutions; ++ : computed solutions. In the last graph, ++ : z and xx : z . (a): the random projection method (3.4). 1

2

15

pressure

density

50

4

45 3.5 40 3

35 30

2.5

25 2

20 15

1.5

10 1 5 0

0

10

20

30

40

0.5

50

0

temperature

10

20

30

40

mass fractions of z and z 1

50

2

14 0.9 12

0.8 0.7

10

0.6 8

0.5 0.4

6

0.3 4

0.2 0.1

2

0 0

0

10

20

30

40

−0.1 30

50

Figure 5.1 (cont'd). (b): the deterministic method.

16

35

40

45

50

pressure

density

100

3.5

80

3

60

2.5

40

2

20

1.5

0

1 0

10

20

30

40

50

0

temperature

10

20

30

40

mass fractions of z and z 1

45

50

2

0.7

40

0.6

35 0.5 30 0.4

25

0.3

20 15

0.2

10 0.1 5 0

0 −5

0

10

20

30

40

−0.1 35

50

40

45

Figure 5.2: Numerical solutions of Example 5.2 at t = 3:0 calculated by the random projection method (3.4) with h = 0:25, k = 0:01. { : `exact' solutions; ++ : computed solutions. In the last graph, ++ : z and xx :z . 1

2

17

pressure

density 3.5

90 80

3

70 2.5

60 50

2 40 30

1.5

20 1

10 0 0

10

20

30

40

0.5

50

0

10

20

30

40

mass fractions of z ; z and z

temperature

1

35

0.7

30

0.6

25

0.5

2

50

3

0.4

20

0.3 15 0.2 10 0.1 5 0 0 0

10

20

30

40

−0.1 30

50

32

34

36

38

40

Figure 5.3: Numerical solutions of Example 5.3 at t = 3:0 calculated by the random projection method (3.4) with h = 0:25, k = 0:01. { : `exact' solutions; ++ : computed solutions. In the last graph, ++ : z ; xx : z and oo : z . 1

2

18

3

pressure

density

140 4 120 3.5 100 3 80 2.5 60 2 40 1.5

20

1

0 0

10

20

30

40

50

0

10

20

30

40

mass fractions of z ; z ; z and z

temperature

1

40

0.8

35

0.7

30

0.6

2

3

50

4

0.5

25

0.4

20

0.3

15

0.2 10 0.1 5 0 0 0

10

20

30

40

−0.1 30

50

32

34

36

38

40

Figure 5.4: Numerical solutions of Example 5.4 at t = 3:0 calculated by the random projection method (3.4) with h = 0:25, k = 0:01. { : `exact' solutions; ++ : computed solutions. In the last graph, ++ : z ; xx : z ; oo : z and ** : z . 1

2

19

3

4

t=1

t=2

50

50

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

5

0

0 0

10

20

30

40

50

0

10

20

t=4 50

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

5 0

10

20

40

50

40

50

t=6

50

0

30

30

40

0

50

0

10

20

30

Figure 5.5 Numerical solutions of Example 5.5 calculated by the 2d random projection method (4.7) with h = 0:5, k = 0:01. (a): velocity elds at dierent times.

20

t=1

t=2 35

40

30 p

p

25

30

20 20

15 100z1

10

100z1

10 T

T

5 0

0 0

10

20

30

40

−5

50

0

10

20

t=4

30

40

50

t=6

35

25

30

p

20

25 p

15

20 15

100z1

T

10 100z1

T 10

5

5 0

0 −5

0

10

20

30

40

−5

50

0

10

20

30

40

50

Figure 5.5 (cont'd). (b): Pro les of pressure p ( - - ), temperature T ( { ), and the mass fraction of the rst species multiplied by 100, 100z ( -
), on the line y = x (x  0:0) at dierent times. 1

21

t = 0:0

t = 1:0

25

t = 2:0

25

25 3.46

20

1.37

20

15

3.342.85

1.12 1.24 1.74

10

1.1

1.55

1.28

1.96 3.623.2

2.51

3.753.48 3.07

2.932.1

15

2.35 2.11 2.72

10

2.38

2.24 3.2

1.86 15

2.65

20

2.97

1.9

1.14

3.34

3.21

3.34

10

1.28 1.83 1.14 1.69

2.65

2.97 3.2

1.98 5

1.49

3.21

5

3.34

3.46 0

0

5

10

15

20

25

0

5

10

15

t = 3:0

20

25 1.39 1.29

2.27 2.18

2.03

2.27 1.88 2.37

20

25

0 15

30

1.1

1.47

20

25

1.37 1.28

2.03

20

1.19

1.39

5

2.08

35

40

45

2.12

55

0 45

1.47 50

t = 6:0

60

65

1.38

1.29

1.27

1.29

20

1.19

15

15 2.15 2.08

2.22

10

1.07

2.15 1.14 1.19

2.24 1.29

2.43 0 70

75

80

1.38 85

5

1.19

90

95

100

0 80

1.2 1.14 1.2

1.19

85

2.16 2.35 2.29

1.27 1.34 1.34

90

95

100

1.13

2.29

1.19

1.19 1.26

1.26 1.06 1.32 1.39

2.22

110

1.19

1.19

1.27 1.41 105

1.06

10

5 2.36

1.06

1.19

2.22

2.29

2.29

1.29

85

1.26

2.29

20

1.2

10 2.24

5

80

1.39 1.32

15 2.29

1.191.1

75

1.26

1.14

1.14

1.1

70

2.22 1.2

2.22

2.24

2.14 2.24

65

25

1.41

1.34

2.36

20

2.05

1.34

1.2

1.19 1.19

2.24

0 60

t = 8:0

25 2.43

70

1.27 1.411.34

1.34

2.23 2.09

1.37

55

1.2

5

t = 7:0

25

1.07 1.2

2.36

1.28

2.03

50

1.07

2.36 1.19

1.75

5

1.14

2.29 2.23

1.09

1.09

10

1.2

1.39 1.29 0 30

40

1.14

2.36

2.22

2.27

2.18

2.59

2.41

10

2.47 2.37

35

1.41 1.34 1.27

1.34

15 2.5

1.2

2.47

10

2.41

15 1.1

30

2.36

1.49 3.55 3.65 3.75 2.47

25

2.16 2.29 2.09

2.47 15

1.14 2.51

t = 5:0

2.12

20

1.2

2.51

2.38

t = 4:0

25 2.08

1.28 2.24

5

0 90

95

100

105

110

115

120

125

Figure 5.6. Numerical solutions of Example 5.6 calculated by the 2d random projection method (4.7) with h = 0:5, k = 0:01. (a): density contours at dierent times.

22

t=2 100

t=4 100

p

80

p

80

60

60

40

40

T

T

20

20 300z1

0

300z1

0 0

10

20

30

40

50

0

20

40

t=6

60

80

t=8 100

p

p

80

80

60

60

40

40 T

T

20

20 300z1

300z1

0

0 0

20

40

60

80

100

120

0

50

100

150

Figure 5.6 (cont'd). (b): Pro les of pressure p ( - - ), temperature T ( { ), and the mass fraction of the rst species multiplied by 300, 300z ( -
), on the line y = 12:5 at dierent times. 1

23