Antony Jameson. Stanford University Aerospace Computing Laboratory Report ACL

Transcription

1 Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-dimensional Viscous Comressible Flow in a Shock Tube Using Entroy and Kinetic Energy Preserving Schemes Antony Jameson Stanford University Aerosace Comuting Laboratory Reort ACL 007 Moana Surfrider, Waikiki March 007

2 Abstract This aer follows u on the author s recent aer The Construction of Discretely Conservative Finite Volume Schemes that also Globally Conserve Energy or Enthaly. In the case of the gas dynamics equations the revious formulation leads to an entroy reserving EP) scheme. It is shown in the resent aer that it is also ossible to construct the flux of a conservative finite volume scheme to roduce a kinetic energy reserving KEP) scheme which exactly satisfies the global conservation law for kinetic energy. A roof is resented for three dimensional discretization on arbitrary grids. Both the EP and KEP schemes have been alied to the direct numerical simulation of one-dimensional viscous flow in a shock tube. The comutations verify that both schemes can be used to simulate flows with shock waves and contact discontinuities without the introduction of any artificial diffusion. The KEP scheme erformed better in the tests.

3 Introduction Stemming from the ioneering work of Godunov [], rocedures for the construction of nonoscillatory shock caturing schemes are by now well established [, 3, 4, 5, 6]. In general they add artificial diffusion either exlicitly ot imlicitly via uwind oerators in order to satisfy total variation diminishing TVD) or local extremum diminishing LED) roerties [7, 8]. However there is a risk that the artificial diffusion may comromise the accuracy of viscous flow simulations. If the discrete scheme can be constructed to satisfy a global energy estimate of some kind, then it should by stable, at least for smooth solutions, without the need for artificial viscosity. The use of energy estimates to establish stability has a long history, and it is discussed in the classical book of Morton and Richtmyer [9]. One route to achieving discrete energy estimates is to use difference oerators which have a skew-symmetric form. Tyically these oerators are derived by slitting the governing equations in a mixture of conservation and quasilinear form [0]. However, in the treatment of comressible flows with shock waves it is beneficial to write the discrete equations in conservation form. According to the theorem of Lax and Wendroff [], this is sufficient to guarantee that the discrete solution satisfies the correct shock jum conditions as long as it converges in the limit as the mesh interval is reduced. In inviscid comressible fluid flow the total energy, consisting of the sum of the internal and kinetic energy, is conserved, not just the kinetic energy. Corresondingly, it follows from the governing equations that entroy is conserved if no shock waves are formed. In a recent aer [] the author showed that a semi-discrete scheme in conservation form can be constructed so that it globally conserves a generalized entroy function in a smooth flow. Such an entroy reserving EP) scheme is obtained by an aroriate formulation of the numerical fluxes across the interfaces between the grid cells. It requires the use of entroy variables in the evaluation of the flux, although the standard conservative variables are udated at each time ste. There is some latitude in the definition of the generalized entroy function hu) of the state vector u. Harten [3] has given conditions such hu) is a convex function, so that the solution cannot become unbounded if hu) remains bounded. Multilying the governing equations for u by t wt = h then roduces the evolution equation u for h, while t wt reresents the entroy variables. Entroy variables have been used by Hughes, Mallet and Franca [4], and also by Gerritsen and Olsson [5], who roosed a non conservative entroy reserving scheme. 3

4 Although a bound on the kinetic energy does not assure a bound on the solution in a comressible flow, correct simulation of the evolution of kinetic energy is a crucial requirement for accurate simulations of turbulence, where there is an energy cascade between the different eddy scales. In the resent work it is shown that the interface fluxes of a semidiscrete conservative scheme can be constructed in an alternative way which assures that the global discrete kinetic energy evolves in a manner that exactly corresonds to the true equation for kinetic energy. There is some latitude in the definition of the fluxes for such a kinetic energy reserving KEP) scheme, rovided that the fluxes for the continuity and momentum equations satisfy a comatibility condition. Section resents the derivation of the entroy reserving EP) and kinetic energy reserving KEP) schemes for the one dimensional gas dynamics equations. In Section 3 the formulation of the KEP scheme is extended to multi-dimensional viscous comressible flow simulations on arbitrary grids. The KEP roerty is again assured by a comatibility condition between the fluxes for the continuity and momentum equations. The roof is comleted by showing that the finite volume scheme satisfies a discrete Gauss theorem for the ressure and stress terms, rovided that these are both evaluated at the cell interfaces by arithmetic averages of their values in the neighboring cells. In Section 4 both the EP and the KEP schemes are alied to the direct numerical simulation DNS) of one dimensional viscous flow in a shock tube. It is demonstrated in numerical exeriments that both schemes can successfully resolve the shock wave, contact discontinuity and exansion fan without adding any artificial diffusion, rovided that a fine enough mesh is used with a number of cells of the order of the Reynolds number. Reresentative numerical calculations include simulations at a Reynolds number in the range based on the seed of sound on meshes with cells. 4

5 Entroy and kinetic energy reserving schemes for the one-dimensional gas dynamics equations This section resents the formulation of fully conservative schemes for one dimensional gas dynamics which are either entroy reserving EP) or kinetic energy reserving KEP). In both cases the global conservation roerty is obtained by roer construction of the interface flux between each air of neighboring cells. The detailed roof for the EP scheme has been given in [], but in order to facilitate the comarison of the EP and KEP schemes it is outlined here Consider the gas dynamics equations in the conservation form Here the state and flux vectors are u = ρ ρv ρe u t + fu) = 0.) x, f = ρv ρv + ρvh.) where ρ is the density, v is the velocity and, E and H are the ressure, energy and enthaly. Also = γ )ρ where γ is the ratio of secific heats. ) E v, H = E + ρ.3) In the absence of shock waves the entroy ) s = log ρ γ.4) is constant, satisfying the advection equation Consider the generalized entroy function s t + v s x = 0.5) hs) = ρgs).6) 5

6 where it has been shown by Harten [3] that h is a convex function of u rovided that / d g dg ds ds < γ.7) Then h satisfies the entroy conservation law t hu) + F u) = 0.8) x where the entroy flux is F = ρvgs).9) Moreover, introducing the entroy variables w T = h u.0) it can be verified that h u f u = F u and hence on multilying.) by w T we recover the entroy conservation law.8) where now the Jacobian matrix f w = f uu w is symmetric. Accordingly f can be exressed as the gradient of a scalar function G, f = G w.) and the entroy flux can be exressed as F = f T w G.) Different choices of the entroy function gs) have been discussed by Harten [3], Hughes, Franca and Mallet [4], and Gerritsen and Olsson [5]. Suose now that.) is aroximated in semi-discrete form on a grid with cell intervals x j, j =, n as where the numerical flux f j+ du j x j dt + f j+ f j = 0.3) is a function of u i over a range of i bracketing j. In order 6

7 to construct an entroy reserving EP) scheme multily.3) by w T and sum by arts to obtain n x j wj T j= du j dt = n j= w T j f j+ f j n = w T f wn T f n+ + j= ) f T j+ w j+ w j ) At interior oints evaluate f T j+ as the mean value of G wj+ in the sense of Roe [4] such that G wj+ w j+ w j ) = G w j+ ) G w j ).4) Also evaluate the boundary fluxes as f = f w ), f n+ = f w n ).5) Then the interior fluxes cancel, and using.0) amd.), we obtain the entroy conservation law in the discrete form n j= x j dh j dt = F w ) F w n ).6) G wj+ can be constructed to satisfy.4) exactly by evaluating it as the integral G wj+ = 0 G w ŵθ)) dθ.7) where ŵθ) = w j + θ w j+ w j ).8) since then G w j+ ) G w j ) = = 0 0 G w ŵθ)) w θ dθ G w ŵθ)) dθ w j+ w j ) Thus we obtain: 7

8 Theorem. The semi-discrete conservation law.3) satisfies the semi-discrete entroy conservation law.6) is the numerical flux is calculated as f j+ = 0 fŵθ)dθ, j =, n where ŵθ) is defined by.8), and the boundary fluxes are defined by.5) The construction of a kinetic energy reserving KEP) scheme requires a different aroach in which the fluxes of the continuity and momentum equations are searately constructed in a comatible manner. Denoting the secific kinetic energy by k, Thus ] k = ρ v, k [ u = v, v, 0 k t = v v ρ ρv) t t = { )} v + ρ v x + v x.9) Suose that the semi-discrete conservation scheme.3) is written searately for the continuity and momentum equations as dρ j x j dt + ρv) j+ ρv) j d x j dt ρv) j + ρv ) j+ ρv ) j + j+ j = 0.0) = 0.) 8

9 Now multilying.0) by v j and.) by v j, adding them and summing by arts, n j= d x j v j dt ρv) j v j = n j= v j n j= = v ρv) n + j= j= ρv j ) j+ ) dρ j = dt v j j+ j n j= ρv j ) j ) d v ) j x j ρ j dt ) n j= + v ρv ) + v + v n { ρv) j+ v j+ v j v j ρv ) j+ ρv ) j ρv) n+ v n ρv ) n+ } ) ρv ) j+ v j+ v j ) ) v n n+ n + j+ v j+ v j ).) Each term in the first sum containing the convective terms can be exanded as { v j+ + v j ρv) j+ } ρv ) j+ v j+ v j ) and will vanish if ρv ) j+ Now evaluating the boundary fluxes as v j+ + v j = ρv) j+.3) ρv) ρv) n+ = ρ v, ρv ) = ρ n v n, ρv ) n+ = ρ v, = ρ n vn, n+ =.4) = n.) reduces to the semi-discrete kinetic energy conservation law n j= v ) j x j ρ j ) ) v vn = v + ρ v n n + ρ n + n j= j+ v j+ + v j ).5) 9

10 Denoting the arithmetic average of any quantity q between j + and j as q = q j+ + q j ) the interface ressure may be evaluated as Also if one sets j+ =.6) ρv) j+ ) ρv j+ = ρ v.7) = ρ v.8) condition.3) is satisfied. Consistently one may set The foregoing argument establishes ρvh) j+ = ρ v H.9) Theorem. The semi-discrete conservation law.3) satisfies the semi-discrete kinetic energy global conservation law.5) if the fluxes for the continuity and momentum equations satisfy condition.3) and the boundary fluxes are calculated by equations.4). Condition.3) allows some latitude in the construction of the fluxes. For examle it is also satisfied if one sets ρv) j+ ) ρv j+ ρvh) j+ = ρv = ρv v = ρv H instead of equations.7).9). 0

11 3 Kinetic energy reserving KEP) scheme for multidimensional viscous flow The extension of the entroy reserving EP) scheme to multi-dimensional gas dynamics on general grids has been given in []. In this section it is shown how to extend the kinetic energy reserving KEP) scheme to multi-dimensional viscous comressible flow. Denote the density, velocity comonents, ressure, energy and enthaly by, v i,, E and H, where the suerscrit i is used to denote the i th coordinate direction. This is convenient because subscrits will be needed to denote the grid location. Reeated suerscrits will be used to indicate summation over the coordinate directions. Also ) The viscous stress tensor is = γ ) ρe vi, H = E + ρ 3.) ) v σ ij i = µ x + vj ij vk + λδ 3.) j x i x k where δ ij is the Kronecker delta, and µ and λ are the viscosity coefficients. Usually λ = 3 µ. The heat flux is q j = κ T x j 3.3) where T is the temerature and κ is the coefficient of heat conduction. equations can now be written in conservation form as The governing u t + x f i u) = 0 3.4) i where the state and flux vectors are u = ρ ρv ρv ρv 3 ρe, f i = ρv i ρv i v σ i + δ i ρv i v σ i + δ i ρv i v 3 σ i3 + δ i3 ρv i H v j σ ij q j 3.5)

12 The kinetic energy is again denoted by k where [ ] k = ρ vi, k u = vi, v, v, v 3, 0 3.6) and the kinetic energy conservation law is k t + ) } {v j + ρ vi + v i σ ij x j = vj vi σij 3.7) xj x j Suose now that the domain is covered by a grid, and the equations are discretized in finite volume form. The discrete variables may be associated either with nodes, each of which is surrounded by a control volume, or with the control volumes themselves. In the first case the control volumes may be taken as dual cells where the nodes are the vertices of the rimary cells. In the second case the control volumes are simly the rimary cells, and the discrete variables may be regarded as cell averages. Each interior control volume, say o, is bounded by faces not necessarily lanar) with a directed face area S o for the face searating the control volumes o and. Each boundary control volume is closed by an outer face with directed area S o which is the negative of the sum S o of the face areas between o and its neighbors. The control volumes may be tetrahedral, hexahedral or of mixed olyhedral form. The semi-discrete finite volume scheme to be considered has the form vol o du o dt + S i of i o = 0 3.8) for every interior control volume, where S i o are the rojected areas of the face S o in the coordinate directions. At a boundary control volume b there is an additional contribution Sb if b i, where the boundary flux vector will be evaluated as f i b = f i u b ) 3.9) Multilying 3.8) by w T o = ) k u o 3.0)

13 and summing over the nodes o vol o dk o dt = o w T o So i fo i b w T b S i b f i b 3.) where the last sum reresents the contribution of the boundaries. Every interior face aears twice in the sums on the right hand side of 3.) with oosite signs for its discrete face area. Thus its contribution is ) w T wo T S i o fo i 3.) Consider now contributions of the convective terms to 3.). these are ) v j vo j S i o ρv i v j) v o j ) vo j S ) o i ρv i o = { ) v j vo j S i ρv i o v j) o Thus the convective contributions of the interior faces will vanish if ρv i ) o v j ) } vo j ρv i v j) o = ρv i ) o v j + v j o) 3.3) Defining the average of any quantity q between its values at o and as q = q + q o ) 3.4) condition 3.3) will be satisfied if one set ρv i ) o = ρ v i 3.5) ρv i v j) o = ρ v i v j 3.6) To comlete the derivation it is now necessary to examine the contributions of the ressure and stress tensor to 3.). These can be exressed as vo T o SoP i o i b v T b S i bp i b 3.7) 3

14 where v = v v v 3, P i = δ i σ i δ i σ i δ i3 σ i3 3.8) Let P i o be evaluated as The first term in 3.7) is now P i o = P i + P i o) = P i 3.9) vo T Po i So i o o v T o PS i o i At every interior control volume o the fluxes v T SoP i o i generated by its neighbors can be associated with o, so the contributions at o can be written as vt o Po i So i + vt S i o Po i Here Su o = 0 because it is the sum of the face areas of a closed volume. Thus one can add vo T Po i Si o to obtain P o it v + v o ) S i o = σ ij o This reresents the finite volume discretization of v j So i + o v i So i vi vj σij xi x i for the control volume o. A boundary control volume b receives the contributions P b it v Sb i from the neighbors while it returns the contributions P it b v b Sb i Pb it v b Sb i 4

15 giving the total P b it v + v b ) S i o P it b = P it b = σ ij b v b v + v b ) S i b + P it b Sb i Pb it v b Sb i ) v j Sb i + v j b Si b + P b v b Sb i Pb it v b Sb i v i S i b + v i bs i b ) σ ij b vj b Si b P v i bs i b Combining this result with the result for the interior control columes, and including the convective terms for the boundaries, we finally obtain the semi-discrete global kinetic energy conservation law o vol o dk o dt = b + o S i o { v i o o vo i o + ρ o ) v i S i o σ ij o v i oσ ij o } ) v j So i 3.0) where the first sum on the right hand side over the boundary control volumes reresent the flux through the boundaries and the second sum is the finite volume discretization of This establishes theorem 3. D ) vi vj σij dv xi x i Theorem 3. The semi-discrete conservation law 3.8) satisfies the semi-discrete global kinetic energy conservation law 3.0) if the interface fluxes satisfy condition 3.3) and the boundary fluxes are evaluated by equation 3.9). Note that the discrete viscous terms o σ ij v j S i o guarantee dissiation of the discrete energy because vj S i o is the consistent discretization of vj x i in the control volume. Then slitting vj x i into its symmetric and anti-symmetric arts 5

16 as v j x = ) v j i x + vi + ) v j i x j x vi i x j the contribution of the viscous terms amounts to { ) ) ) v j µ x + vi v j i x j x + vi v k } + λ i x j x k o Setting λ = µ this is 3 o v j µ x + vi i x ) vk j δij 3 x k The discretization of the viscous terms covered by the roof does not have the most comact ossible stencil. Consequently it might allow an udamed odd-even mode. 6

17 4 Direct numerical solution of one-dimensional viscous flow in a shock tube This section resents the results of numerical exeriments in which both the entroy reserving EP) and the kinetic energy reserving KEP) schemes have been alied to the direct numerical simulation DNS) of one dimensional viscous flow in a shock tube. In an analysis of discrete solution methods for the viscous Burgers equation [] it was established that the EP scheme will satisfy the conditions for a local extremum diminishing LED) scheme if the local cell Reynolds number Re c. Here Re c is defined as v x ν where v is the velocity and ν is the kinematic viscosity. This indicates the need to use a mesh with a number of cells roortional to the global Reynolds number vl, where L is the ν global length scale. The comressible Navier Stokes equations are not amenable to such a simle analysis, but it can still be exected that the number of mesh cells needed to fully resolve shock waves and contact discontinuities will be roortional to the Reynolds number, given that the shock thickness is roortional to the coefficient of viscosity, as has been shown by G.I. Taylor and W.D. Hayes [6, 7]. In the numerical exeriments the viscous stress was calculated at each cell interface by discretizing the velocity and temerature gradient on a comact stencil as ) v x ) T x j+ j+ = x v j+ v j ) = x T j+ T j ) The coefficient of viscosity was calculated as a function of the temerature by Sutherland s law µ = T 3/ T + 0.3) 7

18 taking λ = µ, the viscous stress and heat flux assume the form 3 σ j+ q j+ = 4 ) u 3 µ x ) T = κ x where the coefficient of heat conduction κ was obtained from the Prandtl number P r as The Prandtl number was taken to be.75. κ = µc P r Numerical exeriments were erformed using three different flux formulas j+ j+. Simle averaging: f j+ = f u j+) + f u j )). The entroy reserving EP) scheme: f j+ = where w denote the entroy variables and 0 f ŵθ)) dθ ŵθ) = w j + θ w j+ w j ) 3. The kinetic energy reserving KEP) scheme: ρv) j+ = ρ v ) ρv = ρ v j+ ρvh) j+ = ρ v H In the EP scheme the entroy variables were taken to be w T = h u 8

19 where ) h = ρe s γ+ γ+ = ρ ρ γ Accordingly the entroy variables assume the comaratively simle form w = u 3 u u, u = w 3 w w where = γ ) γ + e s γ γ+ γ+ = γ + γ It was remarked in [] that the energy or entroy reserving roerty could be imaired by the time discretization scheme. One solution to this difficulty is to use an imlicit timesteing scheme of Crank Nicolson tye in which the satial derivatives are evaluated using the average value of the state vectors between the beginning and the end of each time ste, ū j = u n+ j ) + u n j This requires the use of inner iterations in each time ste. In order to avoid this cost, Shu s total variation diminishing TVD) scheme [8] was used for the time integration in all the numerical exeriments. Writing the semi-discrete scheme in the form du dt + Ru) = 0 4.) where Ru) reresents the discretized satial derivative, this advances the solution during one time ste by the three stage scheme u ) = u 0) t Ru 0) ) u ) = 3 4 u0) + 4 u) 4 t Ru) ) u 3) = 3 u0) + 3 u) 3 t Ru) ) The test case for the numerical exeriments was the well known examle originally roosed by Sod [9]. The shock tube extends over the range 0 x, with a discontinuity 9

20 in the initial data at x =.5. The left and right states are L =.0, R =. ρ L =.0, ρ R =.5 v L = 0, v R = 0 The Reynolds number is based on the seed of sound of the left state Re = ρ Lc L µ, c γl L = ρ L The numerical exeriments confirm that the EP and KEP schemes both enable direct numerical simulation DNS) of the viscous flow in the shock tube, rovided that a fine enough mesh is used, while significant oscillations can be observed in the solution when simle flux averaging is used scheme ). It is interesting that the oscillations are rimary observed in the exansion region. As a reresentative examle, Figures show the solutions for a Reynolds number Re = 5000 which were obtained with the three schemes on a grid with 4096 cells, at the time t =.36. The figures also dislay the exact inviscid solution with a solid line.with simle flux averaging there are oscillations in the entroy measured by 0 of the order of.0. With the EP scheme these oscillations are reduced to the ρ γ ρ γ 0 order of.00, while with the KEP scheme they are further reduced to the order of.000. When the Reynolds number is increased to million, and the calculations are erformed on a mesh with cells, it remains true that the only observable oscillations aear in the exansion region, but with reduced amlitude, while the three schemes exhibit the same order of merit. On the other hand the oscillations in the exansion region are amlified as the Reynolds number and number of mesh cells are reduced, as is illustrated in Figures In all cases surrisingly to the author) the KEP scheme erforms better than the EP scheme for this model roblem, while simle flux averaging is inadequate. 0

21 a) Pressure b) Density c) Velocity d) Energy Figure 4.: Simle averaging of the flux: 4096 mesh cells, Reynolds number 5000, Comuted solution values +, Exact inviscid solution

22 a) Pressure b) Density c) Velocity d) Energy Figure 4.: Entroy reserving scheme: 4096 mesh cells, Reynolds number 5000, Comuted solution values +, Exact inviscid solution

23 a) Pressure b) Density c) Velocity d) Energy Figure 4.3: Kinetic energy reserving scheme: 4096 mesh cells, Reynolds number 5000, Comuted solution values +, Exact inviscid solution 3

24 a) Pressure b) Density c) Velocity d) Energy Figure 4.4: Simle averaging of the flux coarse mesh): 5 mesh cells, Reynolds number 500, Comuted solution values +, Exact inviscid solution 4

25 a) Pressure b) Density c) Velocity d) Energy Figure 4.5: Entroy reserving scheme coarse mesh): 5 mesh cells, Reynolds number 500, Comuted solution values +, Exact inviscid solution 5

26 a) Pressure b) Density c) Velocity d) Energy Figure 4.6: Kinetic energy reserving scheme coarse mesh): 5 mesh cells, Reynolds number 500, Comuted solution values +, Exact inviscid solution 6

27 5 Conclusion As a sequel to [], the derivations in this aer establish that it is ossible to construct semi-discrete aroximations to the comressible Navier Stokes equations in conservation form which also discretely reserve the conservation of either entroy the EP scheme) or kinetic energy the KEP scheme). Both these schemes enable the direct numerical simulation of one dimensional viscous flow in a shock tube, rovided that the number of cells in the comutational mesh is of the order of the Reynolds number. The erformance of both the EP and the KEP schemes imroves as the Reynolds number and the number of mesh cells are simultaneously increased. For the model roblem examined in this aer, one-dimensional viscous flow in a shock tube, the KEP scheme erforms better than the EP scheme. The Kolmogoroff scale for the small eddies that can ersist in a viscous turbulent flow is of the order of Re 3 4. Accordingly it aears that by using a mesh with the order of Re 3 cells, direct numerical simulation DNS) of viscous turbulent flow with shock waves will be feasible in the future for high Reynolds number flows. Current high-end comuters attain comuting seeds of the order of 00 teraflos 0 4 floating oint oerations/second). This is about million times faster than high-end comuters 5 years ago. A further increase by a factor of million to 0 0 flos could enable DNS of viscous comressible flow at a Reynolds number of million. This is still short of the flight Reynolds numbers of long range transort aircraft in the range of million, but the eventual use of DNS for comressible turbulent flows can clearly be anticiated 7

28 Acknowledgement The author has benefited tremendously from the continuing suort of the Air Force Office of Scientific Research over the last fifteen years, most recently through Grant Number AF- F , under the direction of Dr. Fariba Fahroo. He is also indebted to Nawee Butsuntorn for his hel in rearing this work in L A TEX. 8

29 References [] S. K. Godunov, A Difference Method for the Numerical Calculation of Discontinous Solutions of Hydrodynamic Equations, Math. Sbornik, 47, 959, [] J. P. Boris and D. L. Book, Flux Corrected Transort, SHASTA, A Fluid Transort Algorithm that Works, J. Com. Phys.,, 973, [3] Bram Van Leer, Towards the Ultimate Conservative Difference Scheme, II, Monotomicity and Conservation Combined in a Second Order Scheme, J. Com. Phys., 4, 974, [4] P. L. Roe, Aroximate Reimann Solvers, Parameter Vectors and Difference Scheme, J. Com. Phys., 43, 98, [5] Amiram Harten, High Resolution Schemes for Hyerbolic Conservation Laws, J. Com. Phys., 49, 983, [6] M. S. Liou and C. J. Steffen, A New Flux Slitting Scheme, J. Com. Phys., 07, 993, 39. [7] Antony Jameson Analysis and Design of Numerical Schemes for Gas Dynamics Artificial Diffusion, Uwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence, International Journal of Comutational Fluid Dynamics, 4, 995, 7 8. [8] Antony Jameson Analysis and Design of Numerical Schemes for Gas Dynamics Artificial Diffusion and Discrete Shock Structure, International Journal of Comutational Fluid Dynamics, 5, 995, 38. [9] R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, Interscience, 967. [0] A.E. Honein and P. Moin, Higher Entroy Conservation and Numerical Stability of Comressible Turbulence Simulations, J. Com. Phys., 0, 004, [] P. D. Lax and B. Wendroff, Systems of Conservation Laws, Comm. Pure. Al. Math., 3, 960,

30 [] A. Jameson, The Construction of Discretely Conservative Finite Volume Schemes that Also Globally Conserve Energy or Entroy, Reort ACL 007, January 007. [3] Amiram Harten, On the Symmetric Form of Systems of Conservation Laws with Entroy, J. Com. Phys., 49, 983, [4] T. J. Hughes, L. P. Franca, and M. Mallet, A New Finite Element Formulation for Comutational Fluid Dynamics: I. Symmetric Forms of the Comressible Euler and Navier Stokes Equations and the Second Law of Thermodynamics, Comuter Methods in Alied Mechanics and Engineering, 54, 986, [5] Margot Gerritsen and Pelle Olsson, Designing an Efficient Solution Strategy for Fluid Flows, J. Com. Phys., 9, 996, [6] G.I. Taylor, The Conditions Necessary for Discontinuous Motion in Gases, Proc. Roy. Soc. London, A84, 90, [7] W.D. Hayes, Gasdynamic Discontinuities, Section D, Fundamentals of Gas Dynamics, edited by Howard W. Emmons, Princeton University Press, ). [8] C. W. Shu, Total-Variation-Diminishing Time Discretizations, SIAM J. Sci. Statist. Comuting, 9, 988, [9] G.A. Sod, Survey of several finite difference methods for systems of nonlinear hyerbolic conservation laws, J. Com. Phys., 6, 978, 3 30