Our setting is a hybrid of two ideas, splitting methods for nonsmooth nonlinear equations [, ] and pseudo-transient continuation (Ψtc) [ 5], a method

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1 Globally Convergent Algorithms for Nonsmooth Nonlinear Equations in Computational Fluid Dynamics? T. Coffey a;, R. J. McMullan b;, C. T. Kelley a;, and D. S. McRae b a North Carolina State University, Center for Research in Scientic Computation and Department of Mathematics, Box 85, Raleigh, N. C , USA b North Carolina State University, Department of Mechanical &Aerospace Engineering Box 79, Raleigh, NC Abstract In this paper we report on a computational study in which a nonsmooth discretization of the Euler equations for flow in a nozzle is solved with splitting method which is in turn globalized with the method of pseudo-transient continuation. Key words: Pseudo-transient continuation, nonlinear equations, steady-state solutions, global convergence, splitting methods, Euler equations, MUSCL approximation Introduction In this paper we put an approach for solving steady-state problems in compressible flow into the context of a global splitting method for a nonsmooth nonlinear equation. We use this paradigm to make a conjecture about a convergence theorem and illustrate the ideas with a numerical experiment.? Version of 6 November. Captain, U S Air Force. Participation funded by the Air Force Institute of Technology, Wright- Patterson AFB, OH. This research was supported in part by National Science Foundation grant DMS-764. Computing activity was partially supported by an allocation from the North Carolina Supercomputing Center. Preprint submitted to Elsevier Science 6 November

2 Our setting is a hybrid of two ideas, splitting methods for nonsmooth nonlinear equations [, ] and pseudo-transient continuation (Ψtc) [ 5], a method for nding steady-state solutions to partial differential equations that often succeeds when methods such as line searches fail. While the theory for Ψtc [] requires smooth nonlinearities, equations with nonsmooth nonlinearities have been solved by using Jacobian data for related smooth problems [, 6 8]. Even in the smooth case [9] Jacobians for simpler discretizations have been used when the nonlinearity is complicated. This is exactly the paradigm of splitting methods. In this introductory section we begin by describing the general structure of our problem. We then give a general discussion of splitting methods and Ψtc, presenting some known theoretical results, and pointing out the gaps between the theory and the problems we consider here. In x we describe the hybrid algorithm which we will study in the remainder of the paper. We state a conjecture about the nature of the nonsmoothness in the application and then state and prove a local convergence result based on that conjecture. A complete analysis will be done in a subsequent paper. In x we describe a sample problem, flow through a nozzle, and show how a second order discretization of the governing Euler equations leads to a nonsmooth nonlinear equation. We show how our hybrid method can be realized with an approximating smooth problem using a rst order discretization. In x 4 we illustrate the ideas with a computational study. In general terms we seek to solve a nonlinear equation of the form R(U) = () where R is the spatial discretization of a problem in a function space. In the case considered here, the steady-state Euler equations, a high-order discretization must incorporate a nonsmooth flux limiter in order to produce an acceptable result []. Because of this nonsmoothness, a conventional Newton's method approachmust be modied. Beyond that, good initial data for a steady-state problem that has complicated structural features such as shocks is difcult to nd, so a nonlinear iteration that performs well for good initial data, like the splitting methods we describe in x. must be modied when the initial data are far from the solution. Even for smooth problems, a conventional strategy, such as a line search [ ] can either stagnate at a point where the Jacobian is singular or, more seriously, converge to a non-physical solution. Ψtc is a way to address this by directly integrating the time-dependent equation to steady state while increasing the time step to retain efciency. While complete temporal accuracy is sacriced with this approach, the quality of the steady state solution is unchanged.

3 . Splitting Methods for Nonsmooth Equations Splitting methods [, ] decompose R in to smooth and nonsmooth ("rough") parts R(U) =R S (U)+R R (U); () where R S is Lipschitz continuously differentiable. This decomposition is used in the Newtonlike iteration U + = U c R S (U c ) R(U c ); () where U c is the current iteration and U + the updated iteration. In [], the assumptions are that Assumption. ffl There is a solution U Λ. ffl RS(U Λ ) is nonsingular. ffl R S is Lipschitz continuously differentiable. ffl R R is Lipschitz continuous. If Assumption. holds, we can relate the splitting iteration () to a Newton iteration for F (U) =R S (U) R S (U Λ ). In fact, U + = U c F (U c ) F (U c )+R S (U c ) (R R (U Λ ) R R (U c )). Therefore, using the well known convergence theory for Newton's method, we have ku + U Λ k = O(kU c U Λ k + fl R ku c U Λ k): (4) If fl R is small, as one assumes in [,4], then the iteration will converge q-linearly. Obtaining a small fl R in practice is difcult [,5]. Fast multi-level methods for parabolic control problems have been based on this idea [5 7], but the splittings must be constructed carefully if the convergence in the terminal phase is to be superlinear. Inexact Newton methods [8] use as a Newton step any s that satises the inexact Newton condition kr S(U c )s + R(U c )k» c kr(u c )k: (5) If, for example, the step s is computed with an iterative method for linear equations, then the forcing term c in (5) is simply the tolerance on the relative linear residual. So, if U + = U c +s, where s satises (5) then (4) becomes ku + U Λ k = O(kU c U Λ k +( c + fl R )ku c U Λ k): (6)

4 . Pseudo-Transient Continuation Ψtc is a method for computing steady-state solutions of discretized time-dependent partial differential equations. Ψtc is effective when standard methods, such as line search methods [ ] fail by either nding non-physical solutions or stagnating at points where the Jacobian is singular. Let R(U) = be the steady state equation corresponding to the time-dependent initial problem du dt = R(U); U() = U : (7) Ψtc begins by integrating accurately in time, closely following the transient behavior in the early stages of the iteration, until an approximate steady state is reached and then increasing the time step, sacricing temporal accuracy in the terminal phase in favor of rapid convergence to steady state. The equation for a Ψtc step is f c I + J c s = R(Uc ); where J c ß R (U c ); (8) where f c is the current time step". One can view (8) as a single step of a Rosenbrock method [9] if f is xed. Ψtc varies the step, but not to maintain temporal accuracy, but rather to move to steady state with the largest possible steps. A common formula, which we use in the numerical experiments in x 4, for adjusting the time step is the switched evolution relaxation" (SER) method, [], f n =min(f kr(u )k=kr(u n )k;f max ): (9) SER is a common approach for fluid flow problems [, 5,, ]. The convergence theory from [] gives conditions under which Ψtc will converge to the desired steady-state solution, which we dene as U Λ =lim t! U(t), where U is the solution of (7). The basic assumptions are that U Λ is a stable steady-state solution and that the approximate Jacobians are well conditioned. Formally: Assumption. ffl R is Lipschitz continuous and uniformly bounded. ffl The initial value problem (7) has a stable steady-state solution U Λ. 4

5 ffl k(i + fr (U)) k»( + f) some, all U near U Λ, f>. ffl R (U Λ ) is nonsingular. Theorem. is a summary of the convergence results from []. In Theorem. we use an inexact Newton method, in which the step satises k(f n I + R (U n ))s n + R(U n )k» n kr(u n )k: () Theorem. Let Assumption. hold. If f is sufciently small, s n satises (), and n» μ < for some μ sufciently small, then f n!, U n! U Λ, and the convergence in the terminal phase of the iteration is described by ku n+ U Λ k + O(( n + f n )ku n U Λ k + ku n U Λ k ) for n sufciently large. In particular, if f max = and n =, the convergence is q-quadratic. The analysis in [] identies three phases in the convergence of the Ψtc iteration. In the early phase, f is small and the effort is spent in resolving those transient effects that must be resolved to nd the solution. At the end of the rst phase, the approximation of the solution is good, and f is increased in the second phase. In the third and nal phase, f is large and the solution is good; this combination leads to fast convergence to a highly accurate solution. Hybrid Algorithm The hybrid algorithm combines splitting and Ψtc by simply using J c = R S (U c )intheequation for the step. (f c I + R S (U c ))s = R(U c ); () and in the inexact Newton condition k(f n I + R S(U n ))s n + R(U n )k» n kr(u n )k: () The resulting method has been used in computational fluid dynamics for many years. In fact, it is the only practical formulation if high-order discretizations are to be used. A complex form of the hybrid was used to generate the results in []. While simple to formulate, there has been no formal analysis of the hybrid. This paper is a rst step in that direction. We perform a computational study of a simple example with a view toward numerical verication of a convergence result. We hope to do a complete analysis in future work. 5

6 We can however, formulate a conjecture about the size of the nonsmooth component and based on that conjecture prove a local convergence result. In the application of interest here, R is the discretization of a partial differential operator. The nonsmoothness comes from methods which are required to make high-order approximations correctly resolve shocks. R S is a low-order discretization of the same operator. For the particular methods in x, R S is rst-order accurate and R is second order accurate, where accuracy in measured in the l norm. This motivates our conjecture. Letting x denote the spatial mesh width, we assume that there is C > such that kr S(U c ) (R R (U Λ ) R R (U c ))k»c ( xku c U Λ k) () for all U c sufciently near U Λ. Here k kwill denote the l norm. We will replace Assumption. with the stronger condition, Assumption. ffl There is a solution U Λ. ffl RS(U Λ ) is nonsingular. ffl R S is Lipschitz continuously differentiable. ffl Equation () holds. Assumption. will allow us to prove a local convergence result. The result will show that not only does U n! U Λ but also f n!, which will imply that the convergence rate is the same as one would expect for an implementation based on (5). This result corresponds to the terminal phase of Ψtc as described in []. As is standard, we state the convergence result by examining the transition between a current iterate U c and anewiterate U +. Theorem. Let Assumption. hold and let» c» μ <. If U c is sufciently near U Λ, μ and x are sufciently small, and f c and f max are sufciently large, then the iteration dened by () and (9) satises ku + U Λ k»c (ku c U Λ k +( c + x + f c )ku c U Λ k); (4) for some C >. Moreover, either f + = f max or f + >f c : (5) Proof. LetU c be near enough to U Λ and let μ and x small enough so that (5) and (6) imply that ffl (4) holds for all» c» μ, 6

7 ffl ku + U Λ k»ku c U Λ k=, and ffl kr(u + )k < kr(u c )k. In that case, (9) implies (5). Assuming that f max > = x,which is certainly the case in realistic applications, Theorem. states that the limiting rate of convergence will be q-linear with a q-factor proportional to x + μ. In the application we describe in x 4, μ = O( x) is roughly the error in an approximate Jacobian. We summarize these statements in a corollary. Corollary. Let Assumption. holds and let» n» μ = O( x) <. If U is sufciently near U Λ, μ and x are sufciently small, f c and f max are sufciently large, then the iteration given by () and (9) converges q-linearly with q-factor proportional to x. Euler Equations for Flow in a Nozzle We consider flow through a nozzle of length L. The nozzle is a surface of revolution about the x axis with cross sectional area S(x). Fig.. Nozzle cross section Direction of Flow S(x) L x The governing equations for Quasi-One Dimensional Euler flow in conservative variables @x [(ρu =; (6) = p ds ; and (7) dx =: (8) 7

8 In (6), ρ(t; x) is density, p(t; x) is pressure, u(t; x) is velocity, S(x) is the cross-sectional area of the flow domain, H = (ρe + p) ρ is stagnation enthalpy, and E(t; x) is total energy. We will express the system in conservation form using the variables U = ρ ρu ρe C A : Let F = ρu ρu + p (ρe + p)u C A and W = p ds dx C A ; (9) where c = q fl p is the speed of sound, and fl is the ratio of the specic heat at constant ρ pressure to the specic heat at constant volume. In this paper the gas is air and fl =:4. The conservation form of the system is SU t +(SF) x = W: () The steady state equation is R(U) =(SF) x W =; <x<l; () where F and W are dened by (9). Using the constitutive laws c E = fl(fl ) + u ; ρh = ρe + p; and p =(fl )(ρe ρu ) ( ideal gas assumption) 8

9 = 6 4 ( fl) u ( fl)u fl (fl )u flue fle (fl )u flu 7 5 : () We will consider two nite volume discretizations. We divide the interval [;L] into N cells of length x = L=N, with cell centers x i and cell edges x i±. All our discretizations have the form R(U) = S i+ F i+ S i F i x 6 4 p i S i+ S i x 7 5 : () Since U, and hence F (U), is given only at the cell centers, one must dene the cell-edge fluxes to fully specify the equations. The denition of the fluxes F i± determine the order of the approximation and the smoothness of the nonlinearity.. Lax-Freidrichs Discretization This is a rst order accurate discretization. We will use it to construct R S algorithm. in the Hybrid The Lax-Friedrichs discretization is bf i+ = (F i+ + F i ) j j i+ (U i+ U i ); bf i = (F i + F i ) j j i (U i U i ); where j j = juj + c. Let br(u) = S i+ bf i+ S i bf i x 6 4 p i S i+ S i x 7 5 : (4) 9

10 Since u 6= in this application, b F is a smooth function of U and, when solving R(U) =, the theory in [] is applicable. The Jacobian b R (U) is tridiagonal. A direct computation of b R (U) requires additional computations of cell-edge values of. Instead we use the approximate Jacobian J(U) = 6 4 d () d () d ( ) d () N d ( ) N d () N 7 5 [N N] ß b R (U): (5) In (5) d ( ) i = S i +(juj + c) i I ; d () i = x S i(juj + c) i I; and d () i = S i+ (juj + c) i+i :. Roe Flux Differencing and MUSCL Extrapolation In this section we describe a second order (in the ` sense) method. The use of a slopelimiter is critical to this high accuracy, but is also the source of the nonsmoothness in the nonlinearity. We dene the cell-edge fluxes through right and left cell-edge values of U. Here U L i+ and U R i+ = U i + 4 = U i+ 4» (»)Φ(r + )(U i i U i ) + ( +»)Φ(r )(U i+ i+ U i )» (»)Φ(r )(U i+ i+ U i+ ) + ( +»)Φ(r + )(U i+ i+ U i ) : (6)

11 In (6), r is dened by the ratio of successive differences of U: r + i = U i+ U i+ U i+ U i ; r i+ = U i U i U i+ U i ; r i+ = U i+ U i U i+ U i+ ; and r + i+ = U i+ U i+ U i+ U i : The nonlinear function Φ is a slope limiter. Φ is nonsmooth and must be used if a higher-order method is to perform well []. Two examples are the minmod limiter Φ M (r) = 8 >< >: min(;r) if r> if r» and the VanLeer limiter Φ VL (r) = r + jrj +r : The cell-edge fluxes are F i+ = F L i+ + F R jaj i+ i+ (U R U i+ L ) ; i+ where F L i+ = F (U L ) and F i+ R i+ = F (U R i+ ): The matrix jaj i+ is another potential source of nonsmoothness. However in this paper the flow is entirely supersonic, and jaj i+ is a smooth function of U. The construction of the Roe-averaged Jacobian A begins by writing the Jacobian () in terms of ρ, u, and H = 6 4 ( fl) u ( fl)u fl (fl )u u(h +(fl ) u ) H +(fl ) u (fl ρ ρ )u flu 7 5 : (7)

12 We then dene weighted variables, R i+ = q ρ i+ ρ i ; ρ i+ = R i+ ρ i ; u i+ = R i+ u i+ +u i ; and H R i+ + i+ = R i+ H i+ +H i : (8) R i+ + The Roe averaged Jacobian is A i+ = 6 4 ( fl) u ( fl)u fl (fl )u u(h +(fl ) u ρ ) H +(fl ) u ρ (fl )u flu 7 5 ; where the variables are evaluated at cell edges. At each cell-edge the Roe-averaged Jacobian A has eigenvalues () = u; () = u + c; () = u c; (9) and eigenvectors r () = u u ; r () = u + c H + uc ρ c ; and r() = u c H uc ρ c : () The matrix jaj is the matrix with the same eigenvectors as A and with fj i jg i= as the eigenvalues. These eigenvalues change sign only if parts of the flow are subsonic and other parts are supersonic, which is not the case in the example considered in this paper. 4 Numerical Results In the computational results reported here, L =, N =, and S(x) = 8 >< >: +4(x ) if :5 <x<:5 otherwise The boundary conditions are supersonic flow in the inlet. The initial iterate is the inlet boundary conditions at all points. To check our numerical results, we computed the exact

13 steady state solution using the isentropic gas equations along with conservation of total pressure and total temperature []. One of the resulting equations is nonlinear and we used Newton's method to determine its solution on each cell. The solutions from Ψtc are qualitatively and quantitatively correct according to this exact steady state solution. We compare two methods, the hybrid and a line search approach which takes the inexact splitting Newton step s = J(U c ) R(U c ) and then lets U + = U c + s where = m where we use the smallest integer m such that kr(u c + s)k»( 4 )kr(u c )k: () For smooth problems there is a complete theory for this approach. Either the iteration will converge to a solution, diverge to innity, or stagnate at a point where the Jacobian is singular []. For splitting Newton methods, one can prove similar results if fl R is sufciently small. For this example, the line search fails by stagnating, as one can see both from the residual history and stepsize curves in Figure. The pressure and density curves are also clearly not correct, lacking the symmetry in the physical solution..5.5 Fig.. Backtracking Line Search Newton: Density.5 x Newton: Pressure x.5.5 x Newton: log(relative residual) Newton: log(step size) Iterations Iterations The hybrid method, where U + = U c (f I + J(U c c )) R(U c ), succeeds. The residuals in Figure converge to zero rapidly in a way consistent with the prediction of Theorem. when μ ß max kj(u n ) RS(U n )k = O( x): n

14 Fig.. Pseudo-transient Continuation PTC: Density 4 x PTC: Pressure x.5.5 x PTC: log(relative residual) PTC: log(step size) Iterations Iterations 5 Conclusion Weshowhow an approach used in the computational fluid dynamics literature for computing steady state solutions of compressible flow problems can be expressed as a splitting method. We prove a local convergence result based on an assumption about the errors in the discretizations and the nonsmooth methods developed in [,]. Numerical results are consistent with the predictions of the theory. References [] X. Chen, T. Yamamoto, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. and Optim. (989) [] M. Heinkenschloß, C. T. Kelley,H.T.Tran, Fast algorithms for nonsmooth compact xed point problems, SIAM J. Numer. Anal. 9 (99) [] C. T. Kelley, D. E. Keyes, Convergence analysis of pseudo-transient continuation, SIAM J. Numer. Anal. 5 (998) [4] D. E. Keyes, M. D. Smooke, A parallelized elliptic solver for reacting flows, in: A. K. Noor (Ed.), Parallel Computations and Their Impact on Mechanics, ASME, 987, pp [5] P. D. Orkwis, D. S. McRae, Newton's method solver for the axisymmetric Navier-Stokes equations, AIAA Journal (99)

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