High Order Well-Balanced Schemes and Applications to Non-Equilibrium Flow with Stiff Source Terms

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1 High Order Well-Balanced Schemes and Applications to Non-Equilibrium Flow with Stiff Source Terms Wei Wang, Chi-Wang Shu, H. C. Yee and Björn Sjögreen February 12, 29 Abstract The stiffness of the source terms in modeling non-equilibrium flow problems containing finite-rate chemistry or combustion poses additional numerical difficulties beyond that for soling non-reacting flows. A well-balanced scheme, which can presere certain non-triial steady state solutions eactly, may help minimize some of these difficulties. In this paper, a simple one dimensional non-equilibrium model with one temperature is considered. We first describe a general strategy to design high order well-balanced finite difference schemes and then study the well-balanced properties of the high order finite difference weighted essentially non-oscillatory (WENO) scheme, modified balanced WENO schemes and arious TVD schemes. The adantages of using a well-balanced scheme in presering steady states and in resoling small perturbations of such states will be shown. Numerical eamples containing both smooth and discontinuous solutions are included to erify the improed accuracy, in addition to the wellbalanced behaior. Key words: Well-balanced schemes, non-equilibrium flow, stiff source terms, chemical reactions, WENO schemes, TVD schemes, nozzle flow. 1 Introduction In the modeling of unsteady reactie problems, the interaction of turbulence with finite-rate chemistry introduces a wide range of space and time scales, leading to additional numerical difficulties. A main difficulty stems from the fact that most numerical algorithms used in reacting flows were originally designed to sole non-reacting fluids. As a result, spatial stiffness due to the reacting source terms and turbulence/chemistry interaction are major stumbling blocks to numerical algorithm deelopment. One of the important numerical issues is the improper numerical treatment of a system of highly coupled stiff non-linear source terms, which will result in possible spurious steady state numerical solutions (see Lafon and Yee [14, 15]). It was also shown in Lafon and Yee [14, 15] that arious ways of discretizing the reaction term and initial data can affect the stability of, This paper is an epanded ersion of On well-balanced schemes for non-equilibrium flow with stiff source terms in Stanford CTR Annual Research Briefs 28. Center for Turbulence Research, Stanford Uniersity, Stanford, CA 9435 Diision of Applied Mathematics, Brown Uniersity, Proidence, RI 2912 NASA Ames Research Center, Moffett Field, CA 9435 Lawrence Liermore National Laboratory, Liermore, CA

2 and conergence to, the spurious numerical steady states and/or the eact steady states. Pointwise ealuation of the source terms appears to be the least stable (see [14, 15, 6, 2]). A well-balanced scheme (for time-dependent PDEs), as coined by LeVeque [16], which can presere certain nontriial steady state solutions eactly, may help minimize some of the spurious numerical behaior. Furthermore, well-balanced schemes capture small perturbations of the steady state solutions with high accuracy, thereby making them etremely well suited for computations of turbulent fluctuations on a mainly steady flow field. While general schemes can only resole perturbations at the leel of truncation error with the specific grid, well-balanced schemes can resole much smaller perturbations, usually at 1% or lower of the main steady state flow. Most work about well-balanced schemes deeloped in the literature is for the shallow water equations (e.g. [5, 8, 9, 11, 13, 19, 21, 26, 33]). We follow the work by Xing and Shu in 25 [24]. They deelop a well-balanced high order finite difference weighted essentially non-oscillatory (WENO) scheme for soling the shallow water equations, which is non-oscillatory, well-balanced for still water, and genuinely high order in smooth regions. In [25], they generalize the high order well-balanced WENO scheme to sole a wider class of hyperbolic systems with separable source terms. In this paper, we apply their approach to construct a high order well-balanced WENO scheme for the equations of non-equilibrium flow with reaction terms in one space dimension. Generalizations to multi-dimension can be done following the procedure gien in [24] for the shallow water equations. Work in this area is forthcoming. The one-dimensional hyperbolic system of conseration laws with source terms (also called a balanced law) U t + F (U, ) = S(U, ) (1) is considered, where U is the solution ector, F (U, ) is the conectie flu and S(U, ) is the source term. This balance law admits steady state solutions in which the source term is eactly balanced by the flu gradient. The objectie of well-balanced schemes is to presere eactly some of these steady state solutions. In non-equilibrium flow containing finite-rate chemistry, the source term represents the production of species from chemical reactions. One important property of this type of source term for non-equilibrium flow (which may make the work easier) is that does not appear eplicitly in S(U, ), i.e. S(U, ) S(U). The well-balanced property of arious popular linear and non-linear numerical schemes in the literature is studied in this paper based on a simple one dimensional model with one temperature and three species (O 2, O and N 2 ). This model is obtained by reducing the model by Gnoffo, Gupta & Shinn [4]. The different behaiors of those numerical schemes in presering steady states and in resoling small perturbations of such states will be shown. High order well-balanced WENO schemes are also designed and applied to the one-dimensional model. The procedure of designing well-balanced schemes presented here is alid for any number of species, although the numerical simulations are only performed for the three species model. We will show that for the stationary steady state solutions of the reactie flow, the well-balanced schemes will gie machine round-off errors regardless of the mesh sizes, while the non well-balanced schemes gie truncation errors consistent with the formal order of accuracy for the schemes. Wellbalanced schemes can resole small perturbations of such steady state solutions well with ery coarse meshes, while the non well-balanced schemes would gie spurious structures in the numerical solutions, which will decrease and eentually disappear with a mesh refinement. Our work indicates the adantage of well-balanced schemes: they can be used to resole small perturbations of the steady state solutions, e.g., turbulent fluctuations, using much coarser meshes than that for the non 2

3 well-balanced schemes, thereby saing a lot of CPU time, especially when the number of species increases. Numerical tests also include high-temperature shock tube and nozzle flows. It will be shown that the well-balanced schemes will not lose any accuracy or other good properties such as non-oscillatory shock-capturing for approimating solutions away from the steady state. 2 Goerning equations Assuming no conduction or radiation, the considered non-equilibrium models are a system of hyperbolic conseration laws with source terms, denoted by U t + F (U) = S(U). (2) Here U, F (U) and S(U) are column ectors with m = n s + 2 components where n s is the number of species. U = (ρ 1,..., ρ ns, ρ, ρe ) T, (3) F (U) = (ρ 1,..., ρ ns, ρ 2 + p, ρe + p) T, (4) S(U) = (s 1,..., s ns,, ) T, (5) where ρ s is the density of species s, is the elocity and e is the internal energy per unit mass of the miture. The total density is defined as ρ = n s s=1 ρ s and the pressure p is gien by p = RT n s s=1 ρ s M s, (6) where R is the uniersal gas constant and M s is the molar mass of species s. The temperature T can be found from the total energy n s n s ρe = ρ s e i,s (T ) + ρ s h s ρ2, (7) s=1 where e i,s = C,s T is the internal energy with C,s = 3R/2M s and 5R/2M s for monoatomic species and diatomic species, respectiely, and the enthalpies h s are constants. The source term S(U) describes the chemical reactions occurring in gas flows which result in changes in the amount of mass of each chemical species. We assume there are J reactions of the form s=1 ν 1,jX 1 + ν 2,jX ν n s,jx ns ν 1,jX 1 + ν 2,jX ν ns,jx ns, j = 1,..., J, (8) where ν 1,j and ν 1,j are respectiely the stoichiometric coefficients of the reactants and products of species i in the jth reaction. For non-equilibrium chemistry, the rate of production of species i due to chemical reaction, may be written as s i = M i J j=1 (ν i,j ν i,j) [ n s ( ρs k f,j M s=1 s ) ν s,j n s kb,j s=1 ( ρs M s ) ν ] s,j, i = 1,..., n s. (9) 3

4 For each reaction j, the forward and backward reaction rates, k f,j and k b,j are assumed to be known functions of temperature. Let the Jacobian matri A = F/ U with (a 1,..., a m ) being the eigenalues of A, (a 1,..., a m ) = (,...,, + a, a), (1) where a is the so-called frozen speed of sound. Denote R as the matri whose columns are eigenectors of A (not to be confused with the R in Eq. (6)). Let a l j+1/2, R j+1/2 denote the quantities a l and R ealuated at some symmetric aerage of U j and U j+1, such as Roe s aerage [2]. Define α j+1/2 = R 1 j+1/2 (U j+1 U j ) (11) as the difference of the local characteristic ariables in the direction. In this paper, the considered schemes are the fifth-order finite-difference WENO schemes [1, 22], second-order semi-implicit predictor-corrector TVD (P-C TVD) [3, 17], second-order symmetric [27] and Harten and Yee TVD scheme [29, 28] and second-order MUSCL scheme [28]. Ecept for the P-C TVD, the eplicit TVD high order Runge-Kutta method [23] as well as the pointwise implicit additie Runge-Kutta (ARK) method [12] are used for time discretization (see Appendi A for more details). 3 Well-balanced WENO schemes and linear schemes A well-balanced scheme refers to a scheme that preseres eactly specific steady state solutions of the goerning equations. We will first consider the 1-D scalar balance law i.e., the steady state solution u satisfying u t + f(u, ) = g(u, ), (12) f(u, ) = g(u, ). (13) A linear finite-difference operator D is defined to be one satisfying D(af 1 +bf 2 ) = ad(f 1 )+bd(f 2 ) for constants a, b and arbitrary grid functions f 1 and f 2. A scheme for Eq. (12) is said to be a linear scheme if all the spatial deriaties are approimated by linear finite-difference operators. Xing & Shu [25] proed that under the following two assumptions regarding Eq. (12) and the steady state solution of Eq. (13), linear schemes with certain restrictions are well-balanced schemes. Furthermore, high-order non-linear WENO schemes can be adapted to become wellbalanced schemes. Assumption 1. The considered steady state presering solution u of Eq. (13) satisfies for a known function r(u, ). r(u, ) = constant, (14) 4

5 Assumption 2. The source term g(u,) can be decomposed as g(u, ) = i s i (r(u, ))t i() (15) for a finite number of functions s i and t i. (Here t i is not to be confused with the time t indicated on all preious conseration laws.) We remark that the non-equilibrium flow including nozzle flow studied later in this paper does satisfy these assumptions. A linear scheme applied to Eq. (13) would hae a truncation error D(f(u, )) i s i (r(u, ))D i (t i ()), (16) where D and D i are linear finite-difference operators used to approimate the spatial deriaties. One restriction to the linear schemes is needed: D i = D for all i (17) when applied to the steady state solution. For such linear schemes we hae Proposition 1. For the balance law Eq. (12) with source term Eq. (15), linear schemes with the restriction Eq. (17) for the steady state solutions satisfying (14) are well-balanced schemes. Proof. For the steady state solutions satisfying Eq. (14), the truncation error for such linear schemes with Eq. (17) reduces to D(f(u, )) i ( = D f(u, ) i s i (r(u, ))D(t i ()) ) s i (r(u, ))t i (), where the linearity of D and the fact that r(u, ) is constant for the steady state solution u are used. Note that for such steady state solution u, f(u, ) i s i(r(u, ))t i () is a constant, because ( d f(u, ) ) s i (r(u, ))t i () d i = f(u, ) i s i (r(u, ))t i() = f(u, ) g(u, ) =. Thus, the truncation error is zero for any consistent finite-difference operator D. Therefore, linear schemes with Eq. (17) presere these steady state solutions eactly. Now the high-order non-linear finite-difference WENO schemes are considered in which the nonlinearity comes from the non-linear weights and the smooth indicators. We follow the procedures described in Xing & Shu [24, 25] for the shallow water equations. First, for the situation without 5

6 flu splitting (19) (e.g., the WENO-Roe scheme in [1]), the WENO approimation to f can be written as r f =j c k f k+j = D f (f) j, k= r where r = 3 for the fifth-order WENO approimation and the coefficients c k depend non-linearly on the smoothness indicators inoling the grid function f j r,..., f j+r. The key idea now is to use the finite difference operator D f, and apply it to approimate t i () in the source terms, i.e., t i ( j) r k= r c k t i ( k+j ) = D f (t i ()) j. (18) Clearly, the finite-difference operator D f, obtained from the high-order WENO procedure is a highorder linear approimation to the first deriatie for any grid function. Therefore, the proof for Proposition 1 will be satisfied and we conclude that the high-order finite-difference WENO scheme as stated aboe, without the flu splitting, and with the special handling of the source terms described aboe, maintains eactly the steady state. Net, WENO schemes with a La-Friedrichs flu splitting, such as WENO-LF and WENO-LLF, are considered. The flu f(u, ) is written as a sum of f + (u, ) and f (u, ), defined by f ± (u) = 1 (f(u) ± αu), (19) 2 where α is taken as α = ma u f (u) (see [1, 22] for more details). In order to obtain a well-balanced scheme, the ±αu in the La-Friedrichs flu splitting (19) is replaced with ( ) r(u, ) ±α sgn r(u, ), (2) u where sgn is the sign function with alues +1 or 1. We would need to assume that r(u,) u does not change sign within the local region. The constant α should be suitably adjusted by the size of r(u,) u in order to maintain enough artificial iscosity (see [25] for more details). The framework described for the scalar case can be easily applied to systems (1). For a system with m equations, we would hae m relationships in the form of Eq. (14): r l (U, ) = constant, l = 1,..., m. (21) In Assumption 2, s i could be arbitrary functions of r l (U, ), and s i and t i can be different for different components of the source ector. The characteristic decomposition procedure does not alter the argument presented for the scalar case (see [24]). The modified WENO schemes through the procedure aboe will maintain eactly the steady state and will be called balanced WENO schemes in the paper. Howeer, the modification of the iscosity coefficient α in the La-Friedrichs building block in order to obtain well-balancedness for the steady state may adersely affect stability near strong shocks for solutions far away from the steady state, if Eq. (2) is used to obtain the well-balanced WENO-LF scheme for the system case. Here an equilibrium limiter is introduced, similar to the one 6

7 used in [18], to determine whether the region is near the steady state or far away from the steady state. The La-Friedrichs flu splitting (19) is changed to with λ := ( ma min min ( 1, ( 1, f ± (u) = 1 (f(u) ± αλu), (22) 2 ( r 1(U i+1, i+1) r 1(U i, i) + r 1(U i 1, i 1) r 1(U i, i) ) 2 r 1(U i+1, i+1) r 1(U i, i) 2 + r 1(U i 1, i 1) r 1(U i, i) 2 +ε ( r m(u i+1, i+1) r m(u i, i) + r m(u i 1, i 1) r m(u i, i) ) 2 r m(u i+1, i+1) r m(u i, i) 2 + r m(u i 1, i 1) r m(u i, i) 2 +ε ),..., )) (23) where ε is a small number to aoid zero in the denominator and we take it as 1 6 in the computation. Near steady state, (23) shows that the differences in r i are close to zero. λ will be near zero when all these differences are small compared with ε. λ returns one in the smooth region if the solution is far from the steady state and then the scheme is the regular WENO-LF scheme. This guarantees that the limiter does not affect the high order accuracy of the scheme in smooth region for general solutions of Eq. (2). In the steady state, since all the r l (l = 1,..., m) are constants, λ returns zero and then the scheme maintains the eact solutions for the steady state. To aoid confusion with balanced WENO schemes mentioned aboe, the WENO-LF with the flu splitting (22) and the equilibrium limiter (23) will be called hybrid WENO-LF. Finally, the well-balanced properties of arious TVD schemes mentioned in Sec. 2 will be inestigated. The semi-implicit Predictor-Corrector TVD scheme [3, 17] for Eq. (2) has the form [1 12 ] ts (U nj ) U (1) j = t ( F n j Fj 1 n ) + ts n j (24) U (1) j = U (1) [ 1 1 ] 2 ts (Uj n ) U (2) j = t Uj n+1 = Uj n + 1 ( U (1) j + U (2) j 2 U (2) j = U (2) j + U (1) ) + j + Uj n (25) ( ) F (1) j+1 F (1) j + tsj n (26) j (27) [ ] R (2) j+1/2 Φ(2) j+1/2 R(2) j 1/2 Φ(2) j 1/2. (28) The third step Eq. (28) acts as a non-linear filter step [31]. The elements of the ector Φ j+1/2, denoted by φ l j+1/2 with l = 1,... m are φ l j+1/2 = 1 2 [ ( ψ(νj+1/2 l ) (νl j+1/2 )2] α l j+1/2 ˆQ ) l j+1/2, (29) where νj+1/2 l = t al j+1/2. (3) The function ψ(z) is an entropy correction to z. One possible form is in [28] { z z δ1 ψ(z) = (z 2 + δ1)/2δ 2, (31) 1 z < δ 1 7

8 where δ 1 is the entropy fi parameter. See [32] for a discussion. ˆQl j+1/2 is an unbiased limiter function which can be with ˆQ l j+1/2 = minmod(αl j 1/2, αl j+1/2 ) + minmod(αl j+1/2, αl j+3/2 ) αl j+1/2 (32) minmod(a, b) = sgn(a) ma{, min[ a, b sgn(a)]}. (33) In this study, only diffusie limiters are considered. If a smooth limiter is preferred, then the minmod function minmod(a, b) is replaced by the following smooth function g(a, b) = [a(b 2 + δ 2 ) + b(a 2 + δ 2 )]/(a 2 + b 2 + 2δ 2 ), (34) where δ 2 is a small parameter between 1 7 to 1 5. The predictor step Eq. (24) and the corrector step Eq. (26) are linear. Howeer, the last filter step is not linear. We will eplore this further in the net section. Note that the accuracy of the scheme (and the two considered TVD and MUSCL schemes) reflects the choice of the ery diffusie limit. Numerical accuracy can be improed with a less diffusie limiter. The numerical flu ˆF j+1/2 for the second-order symmetric TVD scheme [27] is described as where ˆF j+1/2 = 1 2 (F j + F j+1 + R j+1/2 Φ j+1/2 ), (35) φ l j+1/2 = ψ(al j+1/2 )(αl j+1/2 ˆQ l j+1/2 ). (36) Pointwise ealuation to the source term is enough for the accuracy for the symmetric TVD scheme and also for the two TVD schemes in the following. Similar to P-C TVD, the non-linearity of the TVD scheme comes from the ˆQ l j+1/2 part of the numerical flu Eq. (36). The second-order Harten-Yee scheme [29, 28] has the same form as Eq. (35) with φ l j+1/2 = 1 2 ψ(al j+1/2 )(gl j + gl j+1 ) ψ(al j+1/2 + γl j+1/2 )αl j+1/2, (37) where γ l j+1/2 = 1 2 ψ(al j+1/2 ) { (g l j+1 g l j )/αl j+1/2 α l j+1/2 α l j+1/2 =. (38) Eamples of the limiter function g l j can be g l j = minmod(α l j 1/2, αl j+1/2 ) (39) or the smooth Eq. (34). Unlike P-C TVD and the TVD schemes, the second-order MUSCL scheme [28] is fully non-linear. The numerical flu for a MUSCL approach is epressed as ˆF j+1/2 = 1 2 (F (U R j+1/2 ) + F (U L j+1/2 ) ˆR j+1/2 ˆΦj+1/2 ) (4) with U R j+1/2 = U j j+1 (41) 8

9 and U L j+1/2 = U j j. (42) The limiters can be j = minmod(u j+1 U j, U j U j 1 ) (43) or the smooth Eq. (34). We can see that Uj+1/2 R and U j+1/2 L bring non-linearity into eery term of the flu Eq. (4). 4 Numerical study This section presents seeral numerical eamples of two types of flows with non-equilibrium chemistry, i.e., planar flow and nozzle flow. Three test eamples will be performed for each type of flows. The first eample is to numerically erify whether the considered schemes are well-balanced by time-marching on a nontriial steady state. In this test, the well-balanced schemes which presere the steady state solutions eactly will gie round-off numerical errors. Net, this steady state is subject to small perturbations of different ariables. From the numerical behaior of all the considered schemes, we can obsere the well-balanced schemes showing their adantage in resoling the perturbations while the non well-balanced schemes will gie spurious numerical truncation errors. The third eample is a shock tube problem to test the shock-capturing capability of the considered schemes. We want to demonstrate that well-balanced schemes will not destroy the non-oscillatory shock resolution away from the steady state. 4.1 Three species model In all test cases, a simple model for air inoling 3 species, O 2, O and N 2 (ρ 1 =, ρ 2 = ρ O and ρ 3 = ρ N2 ) is used. The model has reactions: where M is a catalytic particle (any of the species present). From Eq. (9), the source term S(U) can be written as with ω = ( The forward reaction rate is O 2 + M O + O + M, (44) S(U) = (2M 1 ω, M 2 ω,,, ), (45) k f (T ) ρ 2 M 2 k b (T ) ( ρ1 M 1 ) ) 2 ( ρ1 + ρ 2 + ρ ) 3. (46) M 1 M 2 M 3 k f = CT 2 e E/T, (47) where C = m 3 mole 1 s 1 and E = 5975K (The unit m is short for meter, s is for second and k is for kelin). The backward reaction rate is k b = k f /k eq, (48) with k eq = ep(b 1 + b 2 log z + b 3 z + b 4 z 2 + b 5 z 3 ), z = 1/T, (49) 9

10 where the constants b k are found in [4]. We consider the steady state presering case (ρ 1 ) t = (ρ 2 ) t = (ρ 3 ) t = = p = constant S(U) =. (5) Thus, r = S(U) = constant, (51) which is of the form (21). Note that does not appear eplicitly in S(U) which makes the procedure simpler because all the t i () = 1. Thus the finite-difference operators D i mentioned in Eq. (16) are absent. Therefore, as described in Sec. 3, linear schemes and WENO-Roe schemes applied to the steady state solution Eq. (5) for the problem Eq. (2) are well-balanced and maintain the original high-order accuracy. WENO-LF, WENO-LLF schemes with suitable modification as described in the preious section are also well-balanced and maintain the original high-order accuracy Well-balanced test The purpose of the first test problem is to numerically erify whether the considered schemes are well-balanced for the special stationary case Eq. (5) with ρ O = (1 +.2 sin(5π)) kg/m 3, p = 1 5 N/m 2, = m/s, (52) with and ρ N2 obtained by the equilibrium state condition (The unit kg is for kilogram and N is for newton). We consider the air which consists of 21% of oygen and 79% of nitrogen. This can be epressed as ρ O + 2 ρ O 2 = 21 ρ N2 (53) M O M O2 79 M N2 which holds for the equilibrium state. In the equilibrium state, since there is no reaction, the species also satisfy the source term S(U) =. This set of steady state solutions is of the form Eq. (5). Eq. (52) is chosen as the initial condition which is also the eact steady state solution, and the results are obtained by time-accurate time-marching on the steady state. The error is measured to be the difference between the eact solution Eq. (52) and the numerical solution. The error and accuracy at t =.1 (about 2, time steps for N = 16 grid points) are listed in Tables 1 and 2. It shows that WENO-Roe, P-C TVD and TVD schemes are well-balanced schemes because they produce machine round-off errors. Howeer, WENO-LF and MUSCL schemes are not well-balanced. The hybrid WENO-LF scheme as stated in Sec. 3 is also erified to be well-balanced. We remark that the super conergence of the results for WENO-LF and MUSCL is due to the simple form of the steady state solutions. Numerically P-C TVD and TVD schemes hae been shown to be well-balanced for the steady state solution Eq. (5). Een though the non-linear term RΦ in the P-C TVD and TVD schemes is not linear, we will eplain why this part will not destroy the well-balanced property in these schemes. Since they hae similar formulas, we will use the symmetric TVD scheme as the eample. 1

11 N error error order error WENO-Roe WENO-LF Hybrid WENO-LF E E E E E E E E E-16 Table 1: L 1 errors for ρ O by WENO schemes with N uniform grid points. N error error error error order minmod limiter P-C TVD symmetric TVD Harten-Yee TVD MUSCL E E E E E E E E E E E-2 2.6E smooth limiter E E E E E E E E E E E E Table 2: L 1 errors for ρ O by TVD schemes with minmod/smooth limiter with N uniform grid points, δ 1 =. We claim that the function Φ = for this particular steady state problem Eq. (5). This is due to the fact that is equal to zero in the steady state solution. Recalling the eigenalue a in Eq. (1), it is easy to see that only the last two entries a ns+1 and a ns+2 are non-zero. The function ψ is in Eq. (31). If the entropy parameter δ 1 is set to be zero, we will hae ψ(a l ) = a l. Therefore φ 1... φ ns are always zeros. Note that for any δ 1 >, P-C TVD and TVD schemes are not well-balanced. Net, let us consider the factor α l j+1/2 ˆQ l j+1/2 in Eq. (29) or Eq. (36), where αl j+1/2 is gien in Eq. (11). The resulting equations are obtained directly from the system [7] α ns+k = ( p ± aρ )/2a 2 k = 1, 2, (54) where α, and the frozen speed of sound a are ealuated at the Roe aerage at j + 1/2, and p = p j+1 p j, etc. Since the pressure p is constant and elocity is zero in the steady state solution, α ns+1 and α ns+2 are eactly zeros. Hence, the non-linear term RΦ is zero and then the P-C TVD and TVD schemes become linear schemes in the steady state solution Eq. (5). By Proposition 1, they are well-balanced schemes towards the steady state solution (5) Small perturbation The following test problems will demonstrate the adantages of well-balanced schemes through the problem of a small perturbation oer a stationary state. 11

12 The same stationary state solution Eq. (52) is considered. First, a small perturbation ɛ = 1 1 sin(π) (about.1% of the mean flow) is added to the density, i.e., ρ O = ρ O + ɛ. (55) The other quantities are kept unperturbed. At t =.1, the difference between the perturbed solutions of density ρ O and the steady state solutions of density ρ O is plotted ( denotes the difference in the figures). The reference results are computed by WENO-Roe with 12 points and are considered to be eact. To improe the iewing, a factor of 1 1 is added to all the figures. The solution of density ρ O by WENO-Roe is depicted in the left subplot of Fig. 1. The adantage of the well-balanced property of WENO-Roe is clearly demonstrated with only 1 points to resole such small perturbation. Although the solution indicates two small bumps in the density plot, these bumps disappear when the mesh is refined to 2 points. Unlike the well-behaed WENO-Roe, the results by WENO-LF, which is not a well-balanced scheme, behae in a ery oscillatory fashion using 1 grid points (middle subplot of Fig. 1). This is due to the fact that the well-balanced schemes can resole the steady state solution eactly, hence they are able to resole a ery small perturbation. Howeer, a scheme that is not well-balanced can only resole the solution when the mesh is refined enough such that the truncation error of the scheme is much smaller than the perturbation. For eample, when the mesh is refined to 3 points for WENO-LF (middle subplot of Fig. 1), the oscillations disappear and the solution is resoled. The right subplot of Fig. 1 shows the good behaior of the hybrid WENO-LF scheme. It resoles the perturbation perfectly with only 1 points. Net, the numerical results by P-C TVD, TVD and MUSCL schemes are discussed, respectiely. As indicated in Sec , P-C TVD and TVD schemes are well-balanced schemes for both the minmod limiter and the smooth limiter. The numerical results of P-C TVD and TVD schemes with the smooth limiter show ery good agreement with the reference solution (right subplots of Figs. 2, 3 and 4), whereas the MUSCL scheme, which is not well-balanced, ehibits oscillatory behaior (Fig. 5 right) in the same mesh N = 3. Howeer, note that in the left subplots of Figs. 2, 3 and 4, results for P-C TVD and TVD schemes with the minmod limiter ehibit some oscillations. These oscillations do not disappear in the mesh refinement until the mesh is etremely fine. This might be caused by the lack of smoothness of the minmod limiter, which is continuous but not differentiable. For the sake of this, only the smooth limiter will be considered in the following test problems. Net our schemes are tested on a perturbation of elocity and a perturbation of energy, i.e. = + ɛ (56) ρe = ρe + ɛ (57) both with ɛ = 1 3 sin(π). Similar to the density case, the difference of the perturbed solutions and the steady state solutions are plotted. But for the elocity case, since the steady state solution is zero, the elocity plots are the perturbed solution. Figure 6 shows the elocity plot under a small perturbation of elocity (56) by WENO-Roe, WENO-LF and hybrid WENO-LF, respectiely. The results by P-C TVD, Harten-Yee TVD and MUSCL are shown in Fig. 7. Figures 8 and 9 show the energy plot under a small perturbation of energy (57) by three WENO schemes and three second order TVD schemes, respectiely. Again, the well-balanced schemes WENO-Roe (left subplots of Figs. 6 and 8), hybrid WENO-LF (right subplots of Figs. 6 and 8), P-C TVD (left subplots of Figs. 7 12

13 ρo ρ O 1 1 ρo ρ O 1 1 ρ O 1 1 ρo Figure 1: Small perturbation of density results by WENO: ɛ = 1 1 sin(π). Left: WENO-Roe (WENO-Roe 1 points: dash-dot; WENO-Roe 2 points: dotted with symbols); Middle: WENO- LF (WENO-LF 1 points: dash-dot; WENO-LF 3 points: dotted with symbols); right: hybrid WENO-LF (hybrid WENO-LF 1 points: dash-dot). Reference: WENO-Roe 12 points: solid ρo ρ O 1 1 ρo Figure 2: Small perturbation of density results by P-C TVD: ɛ = 1 1 sin(π), δ 1 =. Left: with the minmod limiter; Right: with the smooth limiter (P-C TVD 3 points: dash-dot; WENO-Roe 12 points: solid). 13

14 ρo ρ O 1 1 ρo Figure 3: Small perturbation of density results by symmetric TVD: ɛ = 1 1 sin(π), δ 1 =. Left: with the minmod limiter; Right: with the smooth limiter (symmetric TVD 3 points: dash-dot; WENO-Roe 12 points: solid) ρo ρ O 1 1 ρo Figure 4: Small perturbation of density results by Harten-Yee TVD: ɛ = 1 1 sin(π), δ 1 =. Left: with the minmod limiter; Right: with the smooth limiter (Harten-Yee 3 points: dash-dot; WENO-Roe 12 points: solid). 14

15 ρo ρ O 1 1 ρo Figure 5: Small perturbation of density results by MUSCL scheme: ɛ = 1 1 sin(π), δ 1 =. Left: with the minmod limiter; Right: with the smooth limiter (MUSCL 3 points: dash-dot; WENO-Roe 12 points: solid). and 9), Harten-Yee TVD (middle subplots of Figs. 7 and 9) all show ery good agreements with the reference solutions een in a coarse mesh. The results by regular WENO-LF (middle subplots of Figs. 6 and 8) hae some oscillations in the coarse mesh due to the truncation errors. The results by MUSCL scheme (right subplots of Figs. 7 and 9) hae much bigger oscillations een on a mesh with 6 grid points. (The non well-balanced WENO-LF behaes better than the MUSCL scheme, due to the high order conergence of the former.) Through the eamples of perturbations of density, elocity and energy, we clearly demonstrate the adantages of using well-balanced schemes in capturing the small perturbation of steady states. Nearly well-balanced schemes. Recall in the Sec , when the entropy fi parameter δ 1 is non-zero, the last filter step causes trouble, thus P-C TVD and TVD schemes are not well-balanced schemes any more for the steady state solution (5). Howeer, note that the first linear steps of P-C TVD and TVD schemes are still well-balanced. For the sake of this, we call them nearly well-balanced schemes, in contrast to the non well-balanced MUSCL scheme which is not wellbalanced eerywhere. Here we will show the adantages of nearly well-balanced schemes through the small perturbation problems. We perform the same perturbation tests here, but by the non-zero entropy fi parameter schemes. Figure 1 shows the density plots at t =.1 under a perturbation of density (55) by P-C TVD, Harten-Yee TVD and MUSCL schemes, respectiely. Figures 11 and 12 show the elocity and energy plots under a perturbation of elocity (56) and energy (57), respectiely. The analysis in Sec indicates that non-zero δ 1 introduces truncation errors in the filter step, which makes P-C TVD and TVD schemes not well-balanced any more. The smaller δ 1 is, the smaller truncation errors it will cause, thus the better performance these schemes will hae. In Figs. 1 and 12, the results by P-C TVD and Harten-Yee TVD schemes hae smooth fluctuations, which is due to the truncation error by the entropy fi parameter δ 1. MUSCL does not show big difference between zero δ 1 results (right subplot of Fig. 7) and non-zero δ 1 results (right subplot of Fig. 12) in the perturbation of energy problem (57), because the truncation error caused by δ 1 15

16 Figure 6: Small perturbation of elocity results. Perturbation of elocity ɛ = 1 3 sin(π). Dashdot: left: WENO-Roe 1 points; middle: WENO-LF 1 points; right: hybrid WENO-LF 1 points. Solid: WENO-Roe 12 points Figure 7: Small perturbation of elocity results. Perturbation of elocity ɛ = 1 3 sin(π), δ 1 =. Dash-dot: left: P-C TVD 2 points; middle: Harten-Yee TVD 3 points; right: MUSCL scheme 6 points. Solid: WENO-Roe 12 points. 16

17 Figure 8: Small perturbation of energy results. Perturbation of energy ɛ = 1 3 sin(π). Dash-dot: left: WENO-Roe 1 points; middle: WENO-LF 1 points; right: hybrid WENO-LF 1 points. Solid: WENO-Roe 12 points Figure 9: Small perturbation of energy results. Perturbation of energy ɛ = 1 3 sin(π), δ 1 =. Dash-dot: left: P-C TVD 2 points; middle: Harten-Yee TVD 3 points; right: MUSCL scheme 6 points. Solid: WENO-Roe 12 points. 17

18 ρo ρ O ρo ρ O 1 1 ρ O 1 1 ρo Figure 1: Non-zero entropy fi parameter results of density ρ O : perturbation of density ɛ = 1 1 sin(π), δ 1 =.2. Dash-dot: left: P-C TVD 6 points; middle: Harten-Yee TVD 6 points; right: MUSCL scheme 6 points. Solid: WENO-Roe 12 points Figure 11: Non-zero entropy fi parameter results of elocity: perturbation of elocity ɛ = 1 3 sin(π), δ 1 =.2. Dash-dot: left: P-C TVD 2 points; middle: Harten-Yee TVD 3 points; right: MUSCL scheme 6 points. Solid: WENO-Roe 12 points. 18

19 Figure 12: Non-zero entropy fi parameter results of energy: perturbation of energy ɛ = 1 3 sin(π), δ 1 =.2. Dash-dot: left: P-C TVD 6 points; middle: Harten-Yee TVD 6 points; right: MUSCL scheme 6 points. Solid: WENO-Roe 12 points. is much smaller than the original truncation error. MUSCL scheme has error 1 times larger than P-C TVD scheme and Harten-Yee TVD scheme, hence it needs many more points to conerge. The nearly well-balanced P-C TVD scheme and Harten-Yee TVD scheme still perform better than the non well-balanced MUSCL scheme. This can be seen more clearly from a larger perturbation test, for eample, ρe = ρe + ɛ (58) with ɛ = sin(π). The results of energy difference are shown in Fig. 13. In this case, the perturbation is relatiely large compared to the truncation error caused by δ 1. Thus we can see that the nearly well-balanced P-C TVD scheme and Harten-Yee TVD scheme are able to resole this perturbation ery well (left and middle subplots of Fig. 13), whereas the non well-balanced MUSCL scheme cannot do it in the same mesh (right subplot of Fig. 13). Unlike the density and the energy cases, in the perturbation of elocity problem (56), the nonzero δ 1 does not hae any influence on the elocities (Fig. 11). This is because the (n s +1)th element of RΦ is always zero as long as the elocity is zero. Thus, the elocity plots for P-C TVD scheme and Harten-Yee TVD scheme remain as good as the zero δ 1 case (left and middle subplots of Fig. 7) A shock tube problem The third eample consists of a shock tube where the high pressure, high temperature on the left half plane and the low pressure, low temperature on the right plane, both contain air, initially in chemical equilibrium. The conditions are: (p L, T L ) = (1 1 5 N/m 2, 2K), (p R, T R ) = ( N/m 2, 18K), with zero elocity eerywhere and the densities satisfying Eq. (53). Our well-balanced schemes are balanced for the left and right states indiidually, but not through the transition. The WENO-Roe, hybrid WENO-LF, Harten-Yee and MUSCL schemes are tested. We want to demonstrate that the well-balanced (or nearly well-balanced) schemes not only behae nicely near the steady state but also can keep good properties far away from steady state, such in non-oscillatory shock-capturing. 19

20 Figure 13: Non-zero entropy fi parameter results of energy: perturbation of energy ɛ = sin(π), δ 1 =.2. Dash-dot: left: P-C TVD 3 points; middle: Harten-Yee TVD 3 points; right: MUSCL scheme 3 points. Solid: WENO-Roe 12 points p Figure 14: Riemann problem: left: density; middle: elocity; right: pressure (WENO-Roe 3 points: dash line with symbols; WENO-LF 12 points: solid). The reference solution is taken to be regular WENO-LF with 12 points. The results by WENO- Roe, hybrid WENO-LF, Harten-Yee and MUSCL schemes at t =.1 are shown in Figs. 14, 15, 17 and 18, respectiely. For each scheme, the figures of density, elocity and pressure are plotted from left to right. The hybrid WENO-LF gies well resoled, non-oscillatory solutions using 3 uniform cells. We can see clearly from these figures that the well-balanced schemes, i.e., WENO- Roe, hybrid WENO-LF, Harten-Yee, hae the same non-oscillatory shock-capturing capability as the other schemes. 4.2 Nozzle flow In this section, the quasi-one-dimensional non-equilibrium nozzle flow with the reaction terms (see the equilibrium nozzle flow [3]) is considered. The goerning equations for the non-equilibrium flow with 3 species (ρ 1 = O, ρ 2 = O 2, ρ 3 = N 2 ) through a duct of arying cross-section can be written 2

21 p Figure 15: Riemann problem: left: density; middle: elocity; right: pressure (hybrid WENO-LF 3 points: dash line with symbols; WENO-LF 12 points: solid) p Figure 16: Riemann problem: left: density; middle: elocity; right: pressure (P-C TVD 3 points: dash line with symbols; WENO-LF 12 points: solid) p Figure 17: Riemann problem: left: density; middle: elocity; right: pressure (Harten-Yee TVD 3 points: dash line with symbols; WENO-LF 12 points: solid). 21

22 p Figure 18: Riemann problem: left: density; middle: elocity; right: pressure (MUSCL scheme 3 points: dash line with symbols; WENO-LF 12 points: solid). in conseration form as: (ρ i A) t + (ρ i A) = s i A, i = 1,..., 3, (ρa) t + ( (ρ 2 + p)a ) = pa (), (ρe A) t + ((ρe + p)a) =, where A = A() denotes the area of the cross section and s i is gien by Eq. (45). (59) Well-balanced test First the well-balanced properties of our schemes are tested for the same steady state (52), for which the cross section area and the initial conditions are taken as A() = 2 + sin(π), 2, (6) (, ) = m/s, p(, ) = 1 5 N/m 2, T (, ) = 2K, (61) with periodic boundary conditions. The densities of each species are obtained by the equilibrium state condition S(U) = and Eq. (53). The balanced WENO-LF scheme (2) is considered in this section. We take r i (U, ) = ρ i i = 1,..., 3 r 4 (U, ) = r 5 (U, ) = p, (62) which satisfies the Assumption 2 of the well-balanced condition in the steady state (61). Note that an etra source term pa () in addition to the chemical reaction terms appears in Eq. (59). As stated in Sec. 3, this term needs special treatment in order to get the well-balanced WENO schemes. First, pa () is written in the form of (15) with s(r(u, )) = p and t() = A(). Then, it is approimated by the same finite difference operator as that for the flu, i.e., Eq. (18). Table 3 lists the errors and accuracies of the regular WENO schemes and balanced WENO-LF scheme at t =.1 (about 3, time steps for N = 32 points). As before, the eact nontriial steady state solution is known and the error is measured between the eact and numerical 22

23 N error order error order error WENO-Roe WENO-LF Balanced WENO-LF E-7 4.8E E E E E E E E E E E-16 Table 3: L 1 errors of for nozzle flow by WENO schemes with N uniform grid points. N error order error order Harten-Yee MUSCL E-5 1.5E E E E E E E E E Table 4: L 1 errors of for nozzle flow by TVD schemes with N uniform grid points. solutions. As shown in Table 3, none of the regular WENO schemes are well-balanced if just pointwise ealuation is used to the source term. (It is different from the problem in Sec , when there is no spatial deriatie inoled in the source term, the WENO-Roe scheme is well-balanced.) We see a clean fifth order accuracy for the regular WENO schemes. The balanced WENO-LF scheme (2) with special treatment to the source term (18) gies round-off errors, thus it maintains the steady state solution (61) eactly. Table 4 lists the errors and accuracies of Harten-Yee TVD scheme and MUSCL TVD scheme. Harten-Yee TVD scheme appears to be not well-balanced in this case. Recall the analysis in Sec , α ns+k = ( (A()p) ± aa()ρ )/2a 2 k = 1, 2 (63) for the nozzle flow problem Eq. (59). Thus α ns+1 and α ns+2 are no longer zeros. Both Harten-Yee TVD scheme and MUSCL TVD scheme show a second order accuracy for the well-balanced test Small perturbation The following test case is chosen to demonstrate the capability of the proposed scheme for computations on the perturbation of the steady-state solution (61), which cannot be captured well by a non well-balanced scheme. A perturbation of density is considered, i.e., ρ O 2 = + ɛ (64) with ɛ = 1 5 sin(π) (about.1% of the mean flow). The plot of density at the time t =.5 is shown in Fig. 19. The regular WENO-LF scheme with N = 12 points is used as a reference 23

24 WENO-LF 12 balanced WENO-LF 6 Harten-Yee 1 MUSCL Figure 19: Small density perturbation of the nozzle flow. solution. The balanced WENO-LF can capture the perturbation ery well in a mesh size of only 6 points. Howeer, neither Harten-Yee nor MUSCL scheme can do that in a coarse mesh. Similarly, we perturb the elocity = + ɛ (65) with ɛ =.5 sin(π) and the pressure p = p + ɛ (66) with ɛ = 1 sin(π) separately. The well-balanced WENO-LF scheme again presents good resolution of small perturbations in the plots of elocity (Fig. 2) and pressure (Fig. 21). Howeer, the non well-balanced schemes show spurious oscillations and need more points to resole the solution A shock problem Proposed by Anderson in [1], it is concerned with a conergent-diergent nozzle flow with a parabolic area distribution, which is gien by A() = ( 1.5) 2, 3. (67) The initial conditions are taken as the equilibrium state of (, ) = m/s, p(, ) = 1 5 N/m 2, T (, ) = 2K. The boundary conditions are taken as one bar of pressure at the left,.6784 bar of pressure at the right, and 2 K of temperature at both ends. The boundary conditions for the density of each species are obtained from the equilibrium conditions at the boundaries. The flow at boundaries are subsonic for both inflow and outflow. A shock is established inside the pipe. The computation is performed using N = 2 points at t =.1. The pressure p and elocity are plotted. 24

25 WENO-LF 12 balanced WENO-LF 6 Harten-Yee 1 MUSCL Figure 2: Small elocity perturbation of the nozzle flow p WENO-LF 12 balanced WENO-LF 6 Harten-Yee 1 MUSCL Figure 21: Small pressure perturbation of the nozzle flow. 25

26 p p Figure 22: Nozzle flow: (a) left: pressure; (b) right: elocity (balanced WENO-LF 2 points: dash line with symbols; WENO-LF 12 points: solid) p p Figure 23: Nozzle flow: (a) left: pressure; (b) right: elocity (Harten-Yee scheme 2 points: dash line with symbols; WENO-LF 12 points: solid). The results by balanced WENO-LF scheme are shown in Fig. 22. The numerical resolution is ery good without oscillations, erifying the essentially non-oscillatory property of the balanced WENO-LF scheme. For comparison, the results by Harten-Yee TVD and MUSCL are also presented in Figs. 23 and 24, respectiely. 5 Concluding remarks The current results sere as a preliminary study on well-balanced schemes for non-equilibrium flow with source terms. The well-balanced WENO schemes are constructed for the non-equilibrium flow. Numerical eamples are gien to demonstrate the well-balanced property, accuracy, good capturing of the small perturbation to the steady state solutions, and the non-oscillatory shock resolution of the proposed numerical method. Future research will apply the same approach to analyze the wellbalanced properties for the model with larger number of species, and for multi-dimensional flows. 26

27 p p Figure 24: Nozzle flow: (a) left: pressure; (b) right: elocity (MUSCL scheme 2 points: dash line with symbols; WENO-LF 12 points: solid). A more general type of steady state problem with non-zero elocity will also be considered. In this case, the source terms are balanced by the flu gradients. Special attention will be paid to general reactie flows for which perturbation from equilibrium states could be small in some parts of the domain and large in other parts. A Appendi: Additie Runge-Kutta Scheme The implicit time method we are interested in is Additie Runge-Kutta Scheme (ARK) introduced by Kennedy and Carpenter [12]. Rewrite Eq. (2) as { Ut = F N (t, U(t)) + F S (t, U(t)), (68) U() = U, where F N denotes the non-stiff term and F S denotes the stiff term. The ARK scheme to Eq. (68) has the following form i 1 U (i) = U n + t U n+1 = U n + t j= s j= a [N] ij b [N] j F N(t n + c j t, U (j) ) + t F N (t n + c j t, U (j) ) + t i j= s j= a [S] ij F S(t n + c j t, U (j) ), (69) b [S] j F S (t n + c j t, U (j) ), (7) where U () = U n and U (i) approimates U(t n + c i t), i = 1,..., s. The non-stiff and stiff terms are integrated by their own (s + 1)-stage Runge-Kutta methods respectiely. The coefficients a [N] ij, a[s] ij, b[n] j, b [S] j, c j are constrained by order of accuracy and stability considerations. A third-order ARK method are considered in the computation. The coefficients for the 3rd order ARK method (ARK3) we use are list in Table 5. For more details, we refer the readers to [12]. 27

28 a [N] ij i = i = Table 5: The coefficients for ARK3. j = j = 1 j = 2 j = i = a [S] ij i = i = i = b [N] j b [S] j c j Table 6: cfl numbers based on numerical test N WENO 2nd order TVD schemes RK3 ARK3 RK2 IE2 RK3 ARK For the second-order schemes such as Harten-Yee and MUSCL, we use a simple second-order Implicit-eplicit Runge-Kutta scheme (IE2) for the time discretization, which has the form U (1) = U n + tf N (t n, U n ) + tf S (t n, U n ), (71) U n+1 = ((U (1) + U n ) + tf N (t n+1, U (1) ) + tf S (t n+1, U n+1 ))/2. (72) Remark 1. In the steady state or close to steady state problems, the source term is close to zero and thus not stiff. Both an eplicit and an implicit time methods can be used. Howeer, in the state away steady state, such as the shock problem in 4.1.3, using an implicit time method allows a large cfl number and thus saes computational cost. In Table 6, we list the maimum cfl numbers allowed for WENO scheme and second order TVD schemes (Harten-Yee and MUSCL) with different time discretizations mentioned in Appendi A. Implicit methods show big adantage of saing computational cost especially in coarse meshes. When the mesh is refined, the source term becomes less stiff. 28

29 Acknowledgments The authors acknowledge the support of the DOE/SciDAC SAP grant DE-AI2-6ER The work by Björn Sjögreen is performed under the auspices of the U.S. Department of Energy by Lawrence Liermore National Laboratory under Contract DE-AC52-7NA27344 LLNL-JRNL References [1] Anderson, J. D Computational Fluids Dynamics, McGraw-Hill, New York. [2] Engquist, B. & Sjögreen, B Robust difference approimation for stiff iniscid detonation waes, Report CAM 91-5, Dept. of Math., UCLA. [3] Gascón, Ll., & Corberán, J. M. 21 Construction of second-order TVD schemes for nonhomogeneous hyperbolic conseration laws, J. Comp. Phys. 172, [4] Gnoffo, P. A., Gupta, R. N. & Shinn, J. L Conseration equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium, NASA Technical Paper 2867, [5] Greenberg, J. M., & LeRou, A. Y A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33, [6] Griffiths, D. F., Stuart, A. M., & Yee, H. C Numerical wae propagation in hyperbolic problems with nonlinear source terms, SIAM J. Numer. Anal. 29, [7] Grossman, B. & Cinnella, P. 199 Flu-split algorithms for flows with non-equilibrium chemistry and ibrational relaation, J. Comp. Phys. 88, [8] Harten, A., La, P. D., & Van Leer, B On upstream differencing and Goduno-type schemes for hyperbolic conseration laws, SIAM Reiew 25, [9] Hubbard, M. E., & Garcia-Naarro, P. 2 Flu difference splitting and the balancing of source terms and flu gradients, J. Comp. Phys. 165, [1] Jiang, G. & Shu, C.-W Efficient implementation of weighted ENO schemes, J. Comp. Phys. 126, [11] Jin, S. 21 A steady-state capturing method for hyperbolic systems with geometrical source terms, Mathematical Modelling and Numerical Analysis (M 2 AN) 35, [12] Kennedy, C. A. & Carpenter, M. H. 23 Additie Runge-Kutta schemes for conectiondiffusion-reaction equations, Appl. Numer. Math. 44, [13] Kurgano, A., & Ley, D. 22 Central-upwind schemes for the Saint-Venant system, Mathematical Modelling and Numerical Analysis (M 2 AN) 36, [14] Lafon, A. & Yee, H. C Dynamical approach study of spurious steady-state numerical solutions for non-linear differential equations, Part III: The effects of non-linear source terms in reaction-conection equations, Comp. Fluid Dyn. 6,