Simplied Discretization of Systems of Hyperbolic. Conservation Laws Containing Advection Equations. Ronald P. Fedkiw. Barry Merriman.

Transcription

1 Simplied Discretization of Systems of Hyperbolic onservation Laws ontaining dvection Equations Ronald P. Fedkiw arry Merriman Stanley Osher ugust 4, 999 bstract The high speed ow of complex materials can often be modeled by the compressible Euler Equations coupled to (possibly many) additional advection equations. Traditionally, good computational results have been obtained by writing these systems in fully conservative form and applying the general methodology of shock-capturing schemes for systems of hyperbolic conservation laws. In this paper, we show how to obtain the benets of these schemes without the usual complexity of full characteristic decomposition or the restrictions imposed by fully conservative dierencing. Instead, under certain conditions dened in section, the additional advection equations can be discretized individually with a nonconservative scheme while the remaining system is discretized using a fully conservative approach, perhaps based on a characteristic eld decomposition. simple extension of the Lax- Wendro Theorem is presented to show that under certain veriable hypothesis, our nonconservative schemes converge to weak solutions of the fully conservative system. Then this new technique is applied to systems of equations from compressible multiphase ow, chemically reacting ow, and explosive materials modeling. In the last instance, the exibility introduced by this approach is exploited to change a weakly hyperbolic system into an equivalent strictly hyperbolic system, and to remove certain nonphysical modeling assumptions. Research supported in part by ONR N , ONR N

2 Introduction It is widely believed that numerical methods for discretizing the Euler equations should have discrete conservation of mass, momentum, and energy as well as an entropy x for low viscosity schemes. onservative schemes have limit solutions which are weak solutions of the Euler equations yielding accurate representation of the shock speeds and jumps. In contrast, nonconservative schemes generally give limit solutions which have incorrect shock speeds and/or jumps, e.g. see [5]. Note that the work in [3] derived and used special viscosity terms that mimic the Navier-Stokes viscosity in order to reduce these types of nonconservative errors. s models become richer in physics and mathematics, more and more equations are added. Some common examples are the addition of mass fraction equations for chemically reacting ow [7] and the addition of the level set equation for compressible gas ow [8]. nother example is the aer Nunziato (N) two phase model for solid explosives and propellants [], where a second set of Euler equations is added for the solid phase, and a volume fraction equation is added to close the system. In addition, the chemical source terms of the N model require the addition of many more equations, not all of which are known or agreed upon by the community [, ]. s new conservation laws are added to the Euler equations, the Jacobian matrix required for upwind schemes grows accordingly, usually becoming large and unwieldy. For a given system one can often identify a minimal conservative system that contains the truly nonlinear elds. In the chemically reacting ow model and the level set model the Euler equations are the minimal system containing the sound waves. It will be shown that both the mass fraction equations and the level set equation can be written in advection form and upwinded separately according to the particle velocity without degrading the quality of the numerical solution. In fact, the resulting solution is as good, if not better than the fully conservative method. In the N model, the minimal conservative system consists of the two sets of Euler equations which contain the truly nonlinear elds for both the gas sound waves and the solid sound waves. The volume fraction equation and the extra equations added for chemical source terms will be put into advection form and upwinded separately according to the appropriate particle velocity (gas or solid). Note

3 that the N system diers from the rst two examples, where the minimal system was only the Euler equations and all added equations were put into advection form. The reason for this dierence is that the second set of Euler equations, added to model the solid, contains quantities which have jumps that are not advected along streamlines, e.g. shock waves in the solid. One often adds new advection equations of the form to the minimal system. The continuity equation is Z t + uz x = () t + (u) x = () where is the density and u is the velocity. Multiplying equation by and adding it to Z times equation results in (Z) t + (Zu) x = (3) as a new equation in conservation form. If the correct Rankine-Hugoniot jump conditions of the full system are such that discontinuities in Z are advected along streamlines then the technique introduced in this paper applies. This is the case for the mass fraction equations, the level set equation, and the volume fraction equation of the N model, but not the case for the second set of Euler equations in the N model which must be added to the minimal system. s a further illustration, consider S t + us x = (4) where S is the entropy per unit mass. While this equation is valid away from shocks, discontinuities in S are not necessarily advected along streamlines and the technique introduced in this paper does not apply. The advection of discontinuities in Z along streamlines can be made more precise by considering a discontinuity moving with speed D. The well known jump conditions for equations and 3 are and left (u left? D) = right (u right? D) (5) left Z left (u left? D) = right Z right (u right? D) (6) which can be combined to obtain right (Z left? Z right )(u right? D) = (7) 3

4 illustrating that Z must be continuous, unless u right = D = u left. That is, discontinuities of Z move with the particle velocity. Moreover Z is continuous across shocks. Note that these statements are generally not true for the entropy. There are two distinct kinds of advection equations which may be added to a minimal system. The rst kind advects quantities that the minimal system depends on, examples include dependence of the pressure on the mass fractions, the level set function, or the volume fraction. The left and right eigenvectors of the Jacobian matrix are needed in order to use an upwind scheme. These types of advection equations dictate an increase in the size of the left eigenvector, since information is needed from these variables to accurately project into the characteristic elds. However, the right eigenvector remains the length of the minimal system, since only the minimal system is updated using the characteristic elds, while the advection equations are updated individually. The second kind of advection equation advects those quantities that the minimal system does not depend on, even though the new equations may be dependent on the minimal system for their characteristic velocity. These types of equations occur in newer models such as the N equations where the advected quantities have been added in order to model the chemical source terms. These advection equations have no effect on the hyperbolic part of the minimal system and do not change the eigensystem at all, i.e. neither the right nor left eigenvector of the associated Jacobian matrix of the minimal system changes. Here the savings in simplicity of scheme design and programming eort are enormous using our technique rather than standard conservation form discretizations, because one only needs to discretize simple additional advection equations. There are also signicant gains in execution time. In section 5, a conservative equation that introduces a weak hyperbolicity in the N model (which implies that the problem is mildly ill-posed) is discussed. In this instance, the identication of a quantity with discontinuities that advect along streamlines is extremely useful since the conservative equation is equivalent to advecting a quantity which blows up analytically as the reaction proceeds to completion. This problem is easily xed by advecting an equivalent quantity which goes to zero as the reaction proceeds to completion. Furthermore, putting this well behaved advection equation into conservative form and solving in the usual way removes the weak hyperbolicity for the full conservation form as well. Once the advection equations are isolated from the minimal conservative system, greater exibility in scheme design can be exploited. If the advected 4

5 quantity is continuous, e.g. the level set function, then one can use high order essentially nonoscillatory (ENO) Hamilton-Jacobi methods which gain an order of accuracy at extrema over the conservative ENO ux method and are also somewhat easier to implement [, ]. If the advected quantity is discontinuous, our theoretical and numerical results indicate that nonconservative ux form is preferable. That is, we recommend that the spatial term of Z t + uz x = (8) be approximated as uz x u i ^Zi+? ^Z i? 4x (9) where ^Z i+ can be viewed as a numerical ux function consistent with Z. For discontinuous advected quantities, the contact discontinuities can be sharpened with articial compression methods []. In fact, isolating the advection equations in a ux form that allows nonconservative ux dierencing opens up new research avenues such as the pursuit of a fully multidimensional articial compression method, which could not be carried out as easily with conservative numerical methods. 5

6 Theoretical Justication The general theorem concerns the following system of equations in R 3 R + q t + r ~ f (q; Z) = () Z t + ~u rz = () where ~ f = (f; g; h) and ~u = (u; v; w). Note that q, f, g, and h are all ` vectors while Z is an m vector. In regions of smoothness, we assume that equations and are equivalent to a conservative system having ` + m dependent variables and that the jump conditions for the conservative system are [ ~ f ~ N] = D[q] () [Z] ~u ~N? D = (3) where ~N is the unit normal to the surface of discontinuity, and D is the local discontinuity speed in the normal direction. For simplicity of exposition only we consider R R + in the following. We approximate the system using conservation form for the \q" variables and nonconservative ux based form for the \Z" variables. Thus we have q n+ i;j = q n i;j? t x hf ni+ ;j? F ni? ;j + G n i;j+? G n i;j? i (4) where Fi n and ;j Gn i;j are Lipschitz continuous numerical ux functions consistent with f(q; Z) and g(q; Z) respectively. We also have Z n+ i;j = Z n i;j? t x h u n i;j ^Zn i+ ;j? ^Zn i? ;j + v n i;j ^Zi;j+? ^Zi;j? i (5) where ^Zn i and ;j ^Zn i;j are Lipschitz continuous ux functions consistent with Z. We make the following assumptions: Suppose, as t; x; y!, a subsequence of the approximate solutions generated by equations 4 and 5 is bounded a.e. convergent to a piecewise dierentiable limit, (q; Z), for which Z has jumps only where ~u is dierentiable. In addition we assume that 6

7 the rst divided dierences in x and y of these approximations u n i;j and vi;j n are uniformly bounded by an integrable function in any compact set R for which the limit solution (u; v) is dierentiable, i.e. u n i;j and vi;j n are each uniformly bounded in W ; (), and that similar statements are true for Zi;j n in any compact set R for which the limit solution Z is dierentiable. Then we have the following result: Theorem. The piecewise dierentiable limit (q; Z) is a weak solution of equations and, i.e. it is a classical solution when it is dierentiable and it satises equations and 3 at jumps. Remark. Our assumptions are considerably stronger than those of the classical Lax-Wendro theorem [4] which requires only bounded a.e. convergence. Nevertheless, in our calculations in this paper (and elsewhere) using this method, we have observed that our assumptions were valid. In fact the divided dierences of the approximations to ~u were always uniformly bounded away from the discontinuities of the limiting ~u in our calculations, and similarly for Z. Remark. The content of our Theorem is along the lines: convergence plus consistency implies the limit solution satises the original dierential equation. Here, this amounts to showing that the jump conditions in equations and 3 are satised and that equations and are true in regions of smoothness. Remark 3. The Lax-Wendro Theorem states that a converged solution is a weak solution. Thus one has to believe their results have or will converge with further grid renement. Our Theorem requires this as well as the requirement that Z has jumps only where ~u is dierentiable and that the approximations to ~u are uniformly bounded in W ; wherever ~u is dierentiable, with similar requirements on Z and its approximations. One has to believe that no singularities will appear and violate these hypothesis as the grid is rened. This is in the spirit of the Lax-Wendro Theorem, i.e. if it has not happened on the nest computational grid (lack of apparent convergence, or in our case, singularities in the wrong places), then one can quote the appropriate theorem to justify convergence. Remark 4. For an arbitrary system of conservation laws to have the decomposition in equations to 3, it is necessary that one of the eigenvalues of the Jacobian matrix in each dimension be (u; v; w) respectively, each repeated at least m times. This is, of course, only a necessary condition. Remark 5. Our main theorem may appear to contradict some prevailing wisdom. For example in [6] the authors state that the advection equation \does not hold a priori across a shock". This is, of course, generally true, 7

8 but we have proven that it does hold in our sense if Z remains continuous there. Proof. Let '(x; y; t) be a (R R + ) function. We multiply equations 4 and 5 by '(x i ; y j ; t n ) and use the summation-by-parts idea in the proof of the LW theorem, arriving at X i;j;n h txy qi;j n '(x i ; y j ; t n )? '(x i ; y j ; t n? ) t +F n '(xi+ ; y j ; t n )? '(x i ; y j ; t n ) i+ ;j x +G n i;j+ '(xi ; y j+ ; t n )? '(x i ; y j ; t n ) y i! = (6) and X i;j;n h txy Zi;j n! '(x i ; y j ; t n )? '(x i ; y j ; t n? ) t + ^Z u n n i+;j '(x i+ ; y j ; t n )? u n i;j'(x i ; y j ; t n ) i+ ;j x + ^Z n i;j+ v n i;j+ '(x i ; y j+ ; t n )? v n i;j'(x i ; y j ; t n ) y i = (7) We let t; x; y! for the converging subsequence. From equation 6, exactly as in the proof of the Lax-Wendro theorem (which uses the Lebesgue dominated convergence theorem, the Lipschitz continuity of the ux functions, consistency, and the fact that a.e. for bounded measurable functions ), we have Z lim (x + h) = (x) (8) h! R R + q' t + f' x + g' y = (9) 8

9 for any such test function '. This, of course, implies that q is a classical solution of equation wherever q and Z are smooth and that q and Z satisfy equation at jumps. Next we use the identity u n i+;j u n i+;j'(x i+ ; y j ; t n )? u n i;j'(x i ; y j ; t n ) x '(xi+ ; y j ; t n )? '(x i ; y j ; t n ) x = u n + i+;j? u n i;j '(x x i ; y j ; t n ) () and the analogous identity involving \y" divided dierences. hoose ' to have support in a region in which ~u is dierentiable. We let x; y; t! along the converging subsequence. Using equations 7 and, the Lebesgue dominated convergence theorem and the usual Lax-Wendro argument we arrive at Z R R + Z' t + Z (~u r') + Z (r ~u) ' = () If Z is dierentiable in, a simple integration by parts gives us: Z ' (Z t + ~u rz) = () for any such ', hence Z is a classical solution of in. If Z has a jump we take to be a sphere centered at a point of discontinuity. We then let the radius of the sphere go to zero in equation and obtain the jump conditions in equation 3 in a standard fashion. We may prove Z is a classical solution when u jumps with the help of the dominated convergence theorem arriving at equation in a straightforward fashion. 9

10 3 Level Set Equation for ompressible Flow onsider the one dimensional Euler equations with the level set equation in conservation form u u u + E + p = (3) (E + p)u u t where is the density, u is the velocity, E is the total energy per unit volume, p is the pressure, and is the level set function. nother alternative, as suggested in [8], is to replace the conservative level set equation (4th equation) with the level set equation in quasilinear or advection form x t + u x = (4) noting that is valid in advection form since the values of are meant to advect with the particle velocity u, i.e. along streamlines. 3. Projection into haracteristic Fields Modern shock capturing schemes are often based on projection into characteristic elds. Usually the value of the left eigenvector of the Jacobian matrix is frozen and used to locally decompose the system. onsider the Euler equations with the level set equation in advection form. Projection by a left eigenvector, L = (l ; l ; l 3 ; l 4 ), leads to 6 4 (l ; l ; l 3 ; l 4 ) u E t 6 + 4(l ; l ; l 3 ) u u + p (E + p)u x + l 4 u x = (5) where it is obvious that the spatial part of this equation cannot be written in conservation form. In order to obtain conservation form for the projected equation one needs to use the conservative version of the level set equation

11 so that projection by a left eigenvector leads to (l ; l ; l 3 ; l 4 ) u E t (l ; l ; l 3 ; l 4 ) u u + p (E + p)u u x = (6) in conservation form. For this reason, the level set equation is written in conservation form when formulating the Jacobian matrix and the associated eigensystem. 3. onservative Eigensystem The total energy is the sum of the internal energy and the kinetic energy, E = e + u (7) where e is the internal energy per unit mass. The pressure can be written as p = p(; e; ) with partial derivatives p, p e, and p. lternatively, considering the pressure as p = p(; u; E; ) allows one to write the partial derivatives of the pressure with respect to the conserved variables as p = ^H? p ; ^H = c??(h? u ) (8) p (u) =??u; p E =? (9) p () = p (3) where the Gruneisen coecient, the sound speed, and the total mixture enthalpy are dened by s c =? = p e ; p +?p (3) H = E + p (3)

12 The partial derivatives of the pressure are used to obtain the Jacobian matrix?u + ^H? p p u??u??uh + u ^H? up H??u u +?u up?u u (33) and the associated eigensystem ~R = u? c H? uc ; ~ R = u H? b (34) ~R 3 = u + c H + uc ; ~R 4 =? p? (35) ~L b = + u c? p c ;?b u? c ; b ; p c ~L =? b + p c ; b u;?b ;? p c ~L 3 b =? u c? p c ;?b u + c ; b ; p c (36) (37) (38) ~L 4 = (?; ; ; ) (39) where b = p e c ; b = + b u? b H: (4)

13 3.3 Numerical Method When discretizing along the lines of [], the left eigenvectors are used to project into the characteristic elds, the scalar ux in each characteristic eld is discretized based on the associated eigenvalue in that eld, and the right eigenvectors are used to project out of the characteristic elds. Then the contributions from each characteristic eld are added together to produce the total vector ux at a cell interface. These vector uxes are dierenced in the usual manner resulting in a fully conservative numerical method. In order to correctly project into the characteristic elds, the full left eigenvector is always used. However, since conservation form is not required for the last equation, i.e. the level set equation, only the rst three components of the right eigenvectors are used to project out of the characteristic elds. The level set equation is discretized independently in advection form, noting that its characteristic information is determined by u and that nothing about it need be conserved. onsider the Ghost Fluid Method, developed in [5], where p is identically zero. The minimal system no longer depends on the level set equation and only the rst three components of the rst three left and right eigenvectors are needed to update the minimal system. That is, the one dimensional Euler equations can be discretized with a standard conservative scheme while the level set equation is independently discretized in advection form. For both one and two dimensional computational results along with comparisons to the exact solutions, see [5]. 3

14 4 hemically Reacting Flow onsider the two dimensional thermally perfect Euler Equations for multispecies ow with a total of N species, ~U t + [ ~F (~U)] x + [ ~G(~U)] y = ; (4) ~U = u v E Y. Y N? ; ~F (~U) = u u + p uv (E + p)u uy. uy N? ; ~G(~U) = v uv v + p (E + p)v vy. vy N?; (4) E =?p + (u + v ) + NX i= Y i h i! Z T ; h i (T ) = h f i + c p;i (s)ds(43) where t is time, x and y are the spatial dimensions, is the density, u and v are the velocities, E is the energy per unit volume, Y i is the mass fraction of species i, h i is the enthalpy per unit mass of species i, h f i is the heat of formation of species i (enthalpy at K), c p;i is the specic heat at constant pressure of species i, and p is the pressure [7]. Note that Y N = P N?? i= Y i. The pressure is a function of the density, internal energy per unit mass, and the mass fractions, p = p(; e; Y ; ; Y N?), with partial derivatives p ; p e and p Yi where E = e + (u +v ) denes the internal energy per unit mass. The eigenvalues and eigenvectors for the Jacobian matrix of ~F (~U) are obtained by setting = and = in the following formulas, while those for the Jacobian matrix of ~G(~U) are obtained by setting = and =. The eigenvalues are = ^u? c; N+3 = ^u + c (44) = = N+ = ^u: (45) 4

15 where ^u is repeated (N + ) times. The standard construction of upwind dierence schemes requires the full eigensystem of the Jacobian matrix of ~F evaluated as some intermediate value of ~ U. When this Jacobian matrix has a repeated eigenvalue, this involves nding a basis for the associated eigenspace which can be complicated for systems with high multiplicity. The omplementary Projection Method (PM) is an alternative approach where full upwinding is accomplished without the use of the eigenvectors in the repeated eigenvalue eld. Suppose the rst p eigenvalues are repeated, then the corresponding p dimensional characteristic subspace is the span of (~L ; : : :; ~L p ). The remaining left and right eigenvectors can be used to dene f K = ~ L K ~ F (46) and write ~F = ~ F + nx k=p+ f K ~ R K (47) where the vector ~F can be upwinded according to the sign of the repeated eigenvalue = : : : = p, and no basis need be chosen for ~ F. For more details on the PM, see [6]. The PM requires the rst and (N + 3)-rd left and right eigenvectors for the chemically reacting ow example considered here. In addition, since the minimal conservative system is the two dimensional Euler equations, only the rst four entries of the right eigenvectors are needed. The necessary left and right eigenvectors are ~L = b + ^u c + b 3 ;?b u? c ;?b v? c ; b ~L N+3 = b? ^u c + b 3 ;?b u + c ;?b v + c ; b ~R = u? c v? c H? ^uc ; ~ R N+3 = ;?b z ;?b z u + c v + c H + ^uc ; ;?b z N? (48) ; ;?b z N? (49) (5) 5

16 where the unnecessary terms in the right eigenvectors have been omitted for brevity and q = u + v ; ^u = u + v (5) c = r p + pp e ; H = E + p ; (5) b = p e c ; b = + b q? b H; (53) N? X b 3 = b i= Y i z i ; z i =?p Y i p e : (54) 4. Numerical Method The two left eigenvectors are used to project into the two nonlinear elds, while only the rst four entries of the right eigenvectors are used to project out of these elds. The PM is used to construct the complementary subspace ~F = u u + p uv (E + p)u? ~L ~F (~U) ~R? ~L N+3 ~F (~U) ~R N+3 (55) where ~ F has length four and each of its component can be upwind dierenced in the upwind direction determined by u. The resulting ux is combined with the ux contributions from the two nonlinear elds to yield the net numerical ux for the rst four entries of ~F ( ~U). These uxes are dierenced in the usual manner to get the proper upwind conservative discretization for the rst four equations, i.e. for the minimal system. For a dimension by dimension discretization, the same procedure is applied to the ux in the other spatial dimension, and then the total spatial contribution can be used with a TVD Runge Kutta method to update the rst four equations of the 6

17 system, i.e. to update the minimal system. The advection equations for the mass fractions Y. Y N? t + u Y. Y N? x + v Y. Y N?; y = (56) are discretized independently in an equation by equation fashion noting that the characteristic information for the x and y derivatives is determined by u and v respectively and that nothing need be conserved. 4. Examples Several of the examples from [4] and [7] were recomputed using the nonconservative ux based method outlined in this paper. Overall, the calculations using the new technique agreed well with the old calculations. One particularly dicult example from [7] is repeated here. onsider a :m domain with a shock wave located a x = :6m traveling from right to left in a ==7 molar ratio of H =O =r where all gases are assumed to be thermally perfect and a full chemical reaction mechanism is employed as discussed in [7]. The initial data for the shock wave consists of = :7 kg m, u = m kg 3 s and p = 773P a on the left with = :875m, 3 u =?487:34 m s, and p = 35594P a on the right. solid wall boundary condition is applied at x = m and the shock wave reects o this wall changing its direction of travel. s this shock wave passes over the gas a second time, the gas is heated enough to initiate chemical reactions. fter a suitable induction time, a chemical combustion wave forms at the wall and travels to the right eventually overtaking the shock wave resulting in the formation of three waves. From left to right, there is a rarefaction wave, a contact discontinuity, and a detonation wave. Figure shows the solution at 3s with 4 uniform grid cells and 3 equal time steps using the fully conservative ENO-RF method outlined in [7]. Figure shows the same calculation using the nonconservative ux based ENO method (outlined in the appendix) for the mass fraction equations. The PM [6] was used in both calculations. In [7], this was shown to be a very sensitive problem, and the good agreement of the two methods is promising. The only major dierence is in the height of the HO peak, although this seems to have no eect on the rest of the solution. 7

18 grid renement study was carried out with,, 4, 8, and 6 grid cells using 575, 5, 3, 46, and 9 time steps respectively. Figure 3 shows the results with the fully conservative scheme while gure 4 shows the results with the nonconservative ux based method for the mass fraction equations. The position of the lead detonation wave converges with rst order accuracy for both numerical algorithms. In addition, the peaks in the HO mass fraction seem to converge more uniformly (i.e. monotonically) for the nonconservative ux based method. Figures 5, 6, and 7 compare the two methods with, 4, and 6 grid points respectively. The fully conservative method is plotted with x's while the nonconservative ux based method is plotted with o's. 8

19 .6.5 den 5 x vel 3 x 5 press x 5 energy x 3 H mf.5. O mf x 5 HO mf temp O mf.5. OH mf..5. x 5 HO mf gamma.3.5. H mf HO mf r mf Figure : Fully conservative method 9

20 .6.5 den 5 x vel 3 x 5 press x 5 energy x 3 H mf.5. O mf x 5 HO mf temp O mf.5. OH mf..5. x 5 HO mf gamma.3.5. H mf HO mf r mf Figure : Nonconservative ux based algorithm

21 .65 den.6 OH mf HO mf 5 x 5 HO mf Figure 3: Fully conservative method

22 .65 den.6 OH mf HO mf 5 x 5 HO mf Figure 4: Nonconservative ux based algorithm

23 .65 den.6 OH mf HO mf 5 x 5 HO mf Figure 5: grid cells, x-conservative, o-nonconservative 3

24 .65 den.6 OH mf HO mf 5 x 5 HO mf Figure 6: 4 grid cells, x-conservative, o-nonconservative 4

25 .65 den.6 OH mf HO mf 5 x 5 HO mf Figure 7: 6 grid cells, x-conservative, o-nonconservative 5

26 5 N Two Phase Explosives Model onsider a simplied version of the one-dimensional aer-nunziato (N) two phase explosives model [, 3,, ] that ignores nozzling and source terms g g g u g g E g g s s s u s s E s s s t + g u g g ( g u g + p g ) g (E g + p g )u g g s u s s ( s u s + p s ) s (E s + p s )u s s s u s x = (57) where t is time, x is the spatial dimension, g, u g, E g, p g, and g are the density, velocity, energy per unit volume, pressure, and volume fraction of the gas, and s, u s, E s, p s, and s are the density, velocity, energy per unit volume, pressure, and volume fraction of the solid. The seventh equation is the compaction equation which is used to close the system along with the saturation condition, s + g =. ssuming that neither phase has any material strength, the pressure of each phase is a function of the density and internal energy per unit mass of that phase, p g = p g ( g ; e g ) and p s = p s ( s ; e s ), with partial derivatives (p s ) s, (p s ) es, (p g ) g, and (p g ) eg where E g = g e g + gu g and E s = s e s + su s dene the internal energies per unit mass. In addition, assume that the equations of state are dened in a manner consistent with (p g ) g = pg g and (p s ) s = ps s as identities. more detailed version of the model is considered in [8]. Under the above assumptions, the eigenvalues are u g? c g, u g, u g + c g, u s? c s, u s, u s + c s, and u s, with corresponding left and right eigenvectors bg ~L = + u g ;?b gu g? ; b g c g c g ; ; ; ; (58) ~L = (? b g ; b g u g ;?b g ; ; ; ; ) (59) bg ~L 3 =? u g ;?b gu g + ; b g c g c g ; ; ; ; (6) 6

27 ~L 4 = ; ; ; b s + u s c s ;?b su s? c s ; b s ; (6) ~L 5 = (; ; ;? b s ; b s u s ;?b s ; ) (6) ~L 6 = ; ; ; b s? u s c s ;?b su s + c s ; b s ; (63) ~L 7 = (; ; ;?; ; ; s ) (64) ~R = u g? c g H g? u g c g ; ~ R = u g H g? b g ; ~ R3 = u g + c g H g + u g c g (65) ~R 4 = u s? c s H s? u s c s s ; ~R 5 = u s H s? b s s ; ~R 6 = u s + c s H s + u s c s s (66) ~R 7 = s (67)? g = (p g) eg g ; c g = s (p g ) g +? gp g g (68) 7

28 ? s = (p s) es s ; c s = s (p s ) s +? sp s s (69) H g = E g + p g g ; H s = E s + p s s (7) b g =? g ; b c g = + b g u g? b g H g (7) g b s =? s ; b c s = + b s u s? b s H s (7) s 5. Numerical Method The seven left eigenvectors are used to project into the characteristic elds, while only the rst six entries of the seven right eigenvectors are used to project out of these elds, since the minimal system consists of the rst six equations. This avoids the common division by zero problem introduced by the seventh term in the right eigenvectors, i.e. the s term. This term blows up when a chemical reaction depletes s to zero and has forced unphysical modeling, e.g. solid cores [], to avoid the problem. (nother method used to avoid this problem involves the use of central schemes [7, 9].) In [8], a priori knowledge that the seventh entry of the right eigenvectors could be discarded motivated manipulation of the eigensystem in a manner that forces this division by zero problem into that location eliminating it entirely. The volume fraction evolution equation ( s ) t + u s ( s ) x = (73) is an advection equation, and it can be discretized independently noting that the characteristic information is determined by u s and that nothing need be conserved. 8

29 5. Examples In [], a conservation law for the number of particles per unit volume n t + (u s n) x = (74) was added to the one dimensional N model. Inclusion of equation 74 leads to a new weak hyperbolicity when s = due to the \blow-up" of the resulting n s s term in the eigensystem. See [] for details. Using the continuity equation for the solid, equation 74 can be rewritten in advection form n n + u s s = (75) s t s s x where the advected quantity is the number or particles per unit mass. This quantity blows up as the chemical reaction proceeds to completion, i.e. as s!. Since n is dened as the volume fraction divided by the particle volume n = s V p (76) equation 75 is equivalent to + u s V s = (77) p t s V p x which blows up when V p!. The problem with equations 74, 75, and 77 is that they are all derived by advecting a quantity that blows up as the reaction proceeds to completion. This can trivially be avoided by advecting an equivalent quantity that vanishes as the reaction proceeds to completion. s the number of particles per unit mass blows up, the mass per particle vanishes. Replacing equation 75 with ( s V p ) t + u s ( s V p ) x = (78) and solving this equation in advection form removes the weak hyperbolicity. ll that remains are some decoding errors that occur when computing n from equation 76, although n is no longer needed to solve the system. If conservation form is preferred, then equation 78 can be combined with the continuity equation of the solid to obtain? s s V p t +? u s s s V p x = (79) 9

30 which can be substituted in place of equation 74 resulting in an eigensystem n that contains a well behaved s V p term instead of a poorly behaved s s term. In gure 8 we repeat a simple shock tube calculation from [] using equation 78 for the mass per particle to replace the conservation equation for n. We use a m domain with grid cells and a nal time of :6 seconds. Initially, g = s = kg m and p 3 g = p s = : 6 P a on the left, while g = s = kg and m p 3 g = p s = : 5 P a on the right. In addition, u g = u s = m, s particles s = :7, and n = :9 m everywhere. oth 3 the solid and the gas phase are assumerd to obey p = RT and e = c v T with R g = R s = 87 J, kgk (c v) g = 78 J, and kgk (c v) s = 39 J kgk. The results compare well with those from [] where the conservation equation for n was used. Note that the solution for the mass per particle variable only contains a simple contact discontinuity propagating to the right, while the solutions for the particle volume variable and the number of particle per unit volume variable both contain shocks and rarefactions. 3

31 8 6 4 den(g) 5 5 x 5 press(g) den(s) x 5 press(s) x n vel(g) 5 5 temp(g) vel(s) 5 5 temp(s) x Vp x 6 ene(g) phi(g) x 5 ene(s) phi(s) x den(s) * Vp Figure 8: N calculation 3

32 entral Schemes The examples in this paper were discussed based on an upwind discretization with eigenvector projections. In this appendix, a central discretization is briey discussed. See [7] and [9] for examples of central schemes. onsider the one dimensional Euler equations with Z satisfying the following advection equation Z t + uz x = (8) which can be placed into conservation form along with the Euler equations u E Z t + u u + p (E + p)u uz x = (8) and discretized equation by equation with a central scheme. The new technique presented in this paper would not alter the central discretization of the Euler equations, but equation 8 would be discretized in nonconservative form instead of (Z) t + (uz) x = (8) in conservation form. Equation 8 requires articial numerical dissipation proportional to the maximum of ju + cj and ju? cj when discretizing with a central scheme since it includes the continuity equation by construction. Moreover, shocks and rarefactions can occur in Z. In contrast, equation 8 does not require this (possibly large) numerical dissipation and there are no shocks or rarefactions in Z. 3

33 dvection Equation Discretization onsider an advection equation of the form Z t + uz x + vz y = (83) where Z is the advected quantity and u and v are the particle velocities in the x and y direction respectfully. Nonconservative ux based discretizations for Z x are given below. Z y is discretized in a similar fashion and then these two terms are combined with u and v at each grid node before applying a 3rd order TVD Runge Kutta method [9, ] for time integration.. Nonconservative Flux ased ENO Discretization nonconservative extension of the ENO-Roe discretization is used, since there are no nonlinear waves such as shocks and rarefactions [9, ]. The numerical ux function F is dened through the relation where the F i (Z x ) i = F i+? F i? 4x (84) are the values of the numerical ux function at the cell walls. The numerical ux function is constructed at the cell walls by considering the primitive function H of another function h, where h is identical to the numerical ux function F at the cell walls. H is calculated at the cell walls with polynomial interpolation. The zeroth order divided dierences, D, and all higher order even divided i+ dierences of H exist at the cell walls and have the subscript i. The rst order divided dierences Di and all higher order odd divided dierences of H exist at the grid points and have the subscript i. Note that, the zeroth order divided dierences of H vanish with dierentiation and are not needed. The rst order divided dierences of H are, and the higher divided dierences are, D i H = Z i (85) D i+ H = D i+ Z; D i 3 H = 3 D i Z (86) 33

34 onsider a specic grid point i. If u i =, then setting (Z x ) i = will be algorithmically correct since (uz x ) i = is desired, otherwise u i determines the upwind direction. The associated numerical ux function F i+ is dened as follows: If u i >, then k = i. If u i <, then k = i +. Dene Q (x) = (D kh)(x? x i+ ) (87) If jdk? Hj jd Hj, then c = D k+ k? H and k? = k?. Otherwise, c = D H and k? = k. Dene k+ Q (x) = c(x? x k? )(x? x k+ ) (88) If jd 3 k?hj jd3 k? +Hj, then c? = D 3 k?h. Otherwise, c? = D 3 k? +H. Dene Then F i+ = Q 3 (x) = c? (x? x k?? )(x? x k? + )(x? x k? + 3 ) (89) D kh + c ((i? k) + ) 4x + c?? 3(i? k? )? (4x) (9) Likewise, the associated numerical ux function F i? is dened as follows: If u i >, then k = i?. If u i <, then k = i. Dene Q (x) = (D kh)(x? x i? ) (9) If jdk? Hj jd Hj, then c = D k+ k? H and k? = k?. Otherwise, c = D H and k? = k. Dene k+ Q (x) = c(x? x k? )(x? x k+ ) (9) If jd 3 k?hj jd3 k? +Hj, then c? = D 3 k?h. Otherwise, c? = D 3 k? +H. Dene Then F i? = Q 3 (x) = c? (x? x k?? )(x? x k? + )(x? x k? + 3 ) (93) D kh + c ((i?? k) + ) 4x + c?? 3(i?? k? )? (4x) (94) Finally, (Z x ) i is given by equation

35 . Nonconservative Flux ased WENO Discretization onsider a 5th order WENO scheme with the parameter []. Large values of cause the stencil to be biased toward central dierencing (causing oscillations), while small values of cause the stencil to be biased toward 3rd order ENO (lowering the order). To get a stencil biased toward the fth order ux is dened as =?6 maxfv ; v ; v 3; v 4; v 5g +?99 (95) where?99 is used to avoid division by zero and should be much smaller than the rst term in most regions of the domain. onsider a specic grid point i. If u i =, then setting (Z x ) i = will be algorithmically correct since (uz x ) i = is desired, otherwise u i determines the upwind direction. The associated numerical ux function F i+ is dened as follows: If u i >, then v = Z i?, v = Z i?, v 3 = Z i, v 4 = Z i+, and v 5 = Z i+. If u i <, then v = Z i+3, v = Z i+, v 3 = Z i+, v 4 = Z i, and v 5 = Z i?. Dene the smoothness and the weights S = 3 (v? v + v 3 ) + 4 (v? 4v + 3v 3 ) (96) S = 3 (v? v 3 + v 4 ) + 4 (v? v 4 ) (97) S 3 = 3 (v 3? v 4 + v 5 ) + 4 (3v 3? 4v 4 + v 5 ) (98) a = ( + S ) ; w = a a + a + a 3 (99) a = 6 ( + S ) ; w = a a + a + a 3 () a 3 = 3 ( + S 3 ) ; w 3 = a 3 a + a + a 3 () 35

36 to get the ux F i+ = w ( v 3? 7v 6 + v 3 6 ) + w (?v 6 + 5v v 4 3 ) + w 3( v v 4 6? v 5 6 ) () Likewise, the associated numerical ux function F i? is dened as follows: If u i >, then v = Z i?3, v = Z i?, v 3 = Z i?, v 4 = Z i, and v 5 = Z i+. If u i <, then v = Z i+, v = Z i+, v 3 = Z i, v 4 = Z i?, and v 5 = Z i?. Then the smoothness, weights, and ux are dened exactly as above yielding F i? = w ( v 3? 7v 6 + v 3 6 ) + w (?v 6 + 5v v 4 3 ) + w 3( v v 4 6? v 5 6 ) (3) Finally, (Z x ) i = F i +? F i? 4x (4) 36

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