New Streamfunction Approach for Magnetohydrodynamics

Transcription

1 New Streamfunction Approac for Magnetoydrodynamics Kab Seo Kang Brooaven National Laboratory, Computational Science Center, Building 63, Room, Upton NY 973, USA. Summary. We apply te finite element metod to two-dimensional, incompressible MHD, using a streamfunction approac to enforce te divergence-free conditions on te magnetic and velocity fields. Tis problem was considered by Strauss and Longcope [SL9]. In tis paper, we solve te problems wit magnetic and velocity fields instead of te velocity stream function, magnetic flux, and teir derivatives. Considering te multiscale nature of te tilt instability, we study te effect of domain resolution in te tilt instability problem. We use a finite element discretization on unstructured meses and an implicit sceme. We use te PETSc library wit index sets for parallelization. To solve te nonlinear MHD problem, we compare two nonlinear Gauss-Seidel type metods and Newton s metod wit several time step sizes. We use GMRES in PETSc wit multigrid preconditioning to solve te linear subproblems witin te nonlinear solvers. We also study te scalability of tis program on a cluster. MHD and its streamfunction approac Magnetoydrodynamics (MHD) is te fluid dynamics of conducting fluid or plasma, coupled wit Maxwell s equations. Te fluid motion induces currents, wic produce Lorentz body forces on te fluid. Ampere s law relates te currents to te magnetic field. Te MHD approximation is tat te electric field vanises in te moving fluid frame, except for possible resistive effects. In tis study, we consider finite element metods on an unstructured mes for two-dimensional, incompressible MHD, using a streamfunction approac to enforce te divergence-free condition on magnetic and velocity fields and an implicit time difference sceme to allow muc lager time steps. Strauss and Longcope [SL9] applied adaptive finite element metod wit explicit time difference sceme to tis problem. Te incompressible MHD equations are: t B = (v B), v =, B =, t v = v v + ( B) B + µ v, ()

2 Kab Seo Kang were B is te magnetic field, v is te velocity, and µ is te viscosity. To enforce ( incompressibility, ) ( it is) common to introduce stream functions: v = φ y, φ, B = ψ y, ψ. By reformulation for symmetric treatment of te fields, in te sense tat te source function Ω and C are time advanced, and te potentials φ and ψ are obtained at eac time step by solving Poisson equations, we obtain t Ω + [Ω, φ] = [C, ψ] + µ Ω, [ t C + [C, φ] = [Ω, ψ] + φ, ψ φ = Ω, ψ = C, ] [ ] + φ y, ψ y () b y a b were [a, b] = a y. To solve (), we ave to compute te partial derivatives of potentials. Tese partial derivatives can be obtained by solutions of linear problems. To do tis, we ave to introduce four auxiliary variables. It requires te solution of eigt equations at eac step. We use te velocity v and te magnetic field B to reduce ( te number ) of equations to solve (). To tis aim, we put v = (v, v ) = φ y, φ, B = ( ) (B, B ) = ψ y, ψ in equation () and get te following system: Ω t + (v, v ) Ω = (B, B ) C + µ Ω, C t + (v, v ) C = (B, B ) Ω + ([v, B ] + [v, B ]), v = Ω y, v = Ω, B = C y, B = C φ = Ω, ψ = C., (3) In te eigt equations in (3), te last two equations for potentials don t need to be solved to get te solutions in eac time step. If te potentials are needed at a specific time, tey are obtained by solving te last two equations in (3). To solve te Poisson s equations for v and B in (3), we ave to impose boundary conditions wic are compatible to boundary conditions of φ, ψ, Ω, and C. Finite discretization To solve (3), we use te first-order bacward difference (Euler) derivative sceme leading to an implicit sceme wic removes te numerically imposed time-step constraint, allowing muc larger time steps. Tis approac is first order accurate in time and is cosen merely for convenience. Let H denote H (K), H,A denote te subset of H (K) wic satisfy te boundary condition of A, and H,A denote e subspace of H (K) wose elements ave zero values on te Diriclet boundary of A = Ω, v, v, B, B. Multiply te test functions and integrate by parts in eac equation and use te appropriate boundary conditions, to derive te variational form of (3) as

3 New Streamfunction Approac for Magnetoydrodynamics 3 follows: Find X = (Ω, C, v, v, B, B ) H,Ω H H,v H,v H,B H,B suc tat F n (X,Y) = () for all Y = (u, w, p, p, q, q ) H,Ω H H,v H,v H,B H,B, were F n = (F n, F n, F 3, F, F, F 6 ) T, F n(x,y) = Mn t (Ω, u) + (v Ω, u) + µa(ω, u) (B C, u), F n(x,y) = Mn t (C, w) + ( (v C, ) w) (B Ω n, w) P(v,B, w), Ω F 3 (X,Y) = a(v, p ) + y, p, F (X,Y) = a(v, p ) ( Ω, p ), ( ) C F (X,Y) = a(b, q ) + y, q, F 6 (X,Y) = a(b, q ) ( C, q ), (u, w) = K uwdx, Mn t (u, w) = t (u un, w), a(u, w) = K u wdx, P((u, u ), (v, v ), w) = K [u, v ]wdx + K [u, v ]wdx. Let K be a given triangulation of domain K wit te maximum diameter of te element triangles. Let V be te continuous piecewise linear finite element space. Let V A, A = Ω, v, v, B, B, be te subsets of V wic satisfy te boundary conditions of A on every boundary point of K and V A be subspaces of V and H,A. Ten we can write te discretized MHD problems as follows: For eac n, find te solutions X n = (Ωn, Cn, vn,, vn,, Bn,, Bn, ) V Ω V V v V v V B V B for all Y V Ω V V v V v V B V B. on eac discretized times n wic satisfy F n (X n,y ) = () 3 Nonlinear and linear solvers () is a nonlinear problem in te six variables consisting of two time dependent equations and four Poisson equations. However, if we consider te equations separately, eac equation is linear wit respect to one variable. Specifically, te last four equations are linear equations and Poisson problems. From te above observations, we naturally consider a nonlinear Gauss-Seidel iterative solvers (GS) wic solve linear eac equation on one variable in () in consecutive order wit recent approximate solutions. Poisson solvers are well developed and te first two equations are time dependent problems. From tis observation, we consider anoter nonlinear Gauss-Seidel iterative solvers (GS) wic solve first two equations of () as one equation and solve four Poisson equations. Nonlinear Gauss-Seidel iterative metod doesn t guarantee convergence, but converges well in many cases, especially for small time step sizes in time dependent problems. Next, we consider te Newton linearization metod. Newton s metod as, asymptotically, second-order convergence for nonlinear problems and greater

4 Kab Seo Kang scalability wit respect to mes refinement tan te nonlinear Gauss-Seidel metod, but requires computation of te Jacobian of nonlinear problem wic can be complicated. In all tree nonlinear solvers, we need to solve linear problems. Krylov iterative tecniques are suited because tey can be preconditioned for efficiency. Among te various Krylov metods, GMRES (Generalized Minimal RESiduals) is selected because it guarantees convergence wit nonsymmetric, nonpositive definite systems. However, GMRES can be memory intensive (storage increases linearly wit te number of GMRES iterations per Jacobian solve) and expensive (computational complexity of GMRES increases wit te square of te number of GMRES iterations per Jacobian solve). Restarted GMRES can in principal deal wit tese limitations; owever, it lacs a teory of convergence, and stalling is frequently observed in real applications. Preconditioning consists in operating on te system matrix J were J δx = F(x ) (6) wit an operator P (preconditioner) suc tat J P (rigt preconditioning) or P J (left preconditioning) is well-conditioned. In tis study, we use left preconditioning because tis can be implemented easily. Tis is straigtforward to see wen considering te equivalent linear system: P J δx = P F(x ). (7) Notice tat te system in equation (7) is equivalent to te original system (6) for any nonsingular operator P. Tus, te coice of P does not affect te accuracy of te final solution, but crucially determines te rate of convergence of GMRES, and ence te efficiency of te algoritm. In tis study, we use multigrid wic is well nown as a successful preconditioner, as well as a scalable solver in unaccelerated form, for many problems. We consider te symmetrized diagonal term of Jacobian as a reduced system, i.e., J S, = ( JR, + J T ) R,, () were J R, is a bloc diagonal matrix. Te reduced system J S, may be less efficient tan J R, but more numerically stable because it is symmetric and nonsingular. To implement te finite element solver for two-dimensional, incompressible MHD on parallel macines, we use PETSc library wic is well developed for nonlinear PDE problems and easily implements a multigrid preconditioner wit GMRES. We use PETSc s index sets for parallelization of our unstructured finite element discretization.

5 New Streamfunction Approac for Magnetoydrodynamics Numerical experiments : Tilt Instability { [/J ()]J We consider te initial equilibrium state as ψ = (r) y r, r <, (/r r) y r, r >, were J n is te Bessel function of order n, is any constant tat satisfies J () =, and r = x + y (a) t = (b) t = (c) t = (d) t = 7.. Fig.. Contours of Ω, C, φ, and ψ at time t =.,., 6., 7.. In our numerical experiments, we solve on te finite square domain K = [ R, R] [ R, R] wit te initial condition of te tilt instability problem from te above initial equilibrium and perturbation of φ (originating from perturbations of velocity) suc tat { 9.773J (r)y/r if r < Ω() =., C() = {. if r >, φ() = 3 e (x +y ).9966J (r)y/r if r <, ψ() = r)y/r if r >, ( r

6 Kab Seo Kang were = and wit Diriclet boundary conditions Ω(x, y, t) =., φ(x, y, t) =., and ψ(x, y, t) = y y x +y and Neumann boundary condition for C, i.e., C n (x, y, t) =.. Te initial and boundary condition for velocity v and magnetic field B are derived from te initial and boundary condition of Ω, C, φ, and ψ. Numerical simulation results are illustrated in Fig.. Te tilt instability problem is defined on unbounded domain. To investigate te effect of size of domains, we simulate two metods, one uses φ, ψ and its derivatives (SL) and te oter uses v and B (K) on te square domains wit R = and R = 3. Tese numerical simulation results are reported in Fig., te contours of ψ at t = 6.. Te average growt rate γ of inetic energy is sown in Table. Tese numerical simulation results sow tat te solutions of two formulations are closer wen te domain is enlarged, wit te previous approac converging from above and new approac converging from below (a) R = 3 (SL). (b) R = 3(K). (c) R = (SL). (d) R = (K). Fig.. Contours of ψ at T = 7.. Table. Average growt rate γ of inetic energy from t =. to t = 6. previous, R = previous, R = 3 new, R = new, R = From ere, we consider te convergence beaviors of several nonlinear and linear solvers as a function of time step sizes. In Table, we report te number of nonlinear iterations of nonlinear solvers according to time step sizes for te fixed staring time and fixed mes level. We coose t =. and t = 6. as te starting time because many simulations ave trouble at start up time and te magnitudes of te velocity (v, v ) and magnetic field (B, B ) increase wit time. Tese numerical results sow tat GS and Newton metod are more nonlinearly robust tan GS. To investigate anoter convergence beavior, we report te average number of linear iterations in one time step according to preconditioners in Table 3. Numerical results sow tat multigrid preconditioner applying on symmetrized reduced system is robust at t =. and t = 6., but multigrid

7 New Streamfunction Approac for Magnetoydrodynamics 7 Table. Te average number of nonlinear iterations of one time step according to time step sizes dt. dt GS GS NM t = t = 6 t = t = 6 t = t = () * * 7. * * 6 3 applied to te reduced system is robust only at t =., very similar to te symmetrized case, because te values of velocity are small at t =.. Tese results sow tat we ave to use multigrid preconditioner applied to te symmetrized reduced system to get good convergence. Table 3. Te average number of linear iterations in one time step according to time step sizes dt GS(R) GS(S) NM(R) NM(S) t = t = 6 t = t = 6 t = t = 6 t = t = *. *.. *. 7.. * * In Table, we report te average number of nonlinear and linear iterations from t =. to t =. wit dt =. according to te levels. Tese results sow tat two numerical metod GS(S) and Newton metod (S) ave very similar beaviors. Table. Average number of iterations according to te number of level. Solvers GS(S) NM(S) level nonlinear linear nonlinear linear In Table, we report te solution times of one time step according to level and number of processors on te cluster macine BGC (te Brooaven

8 Kab Seo Kang Galaxy Cluster) at BNL. We run te program on te same speed (696 MHz) CPU s. Tis table sows tat Newton s metod ave a better scalabilty GS. Table. Average solving time of one time step (linear system) according to level and number of processors at t =. and dt =. level # CPU GS(S) NM(S) 3.7 (.39).3 (.) 3.3 (.7) 7.76 (.). (.3) 33. (6.6) 9. (7.79). (.3) 3.6 (.) 9.9 (.9) 6. (3.9) 9.9 (.) 6 6. (9.3) 39. (9.6) 3. (36.9) 6.6 (7.) (6.).9 (36.) Conclusions We study te new streamfunction approac metod for two-dimensional, incompressible Magnetoydrodynamics wit finite element metod and tilt instability example. We sow tat nonlinear Gauss-Seidel (GS) and Newton metod ave similar numerical beaviors and ave to employ multigrid preconditioner on te symmetrized reduced system for good linear convergence. Acnowledgement Te autor would lie to tan D. E. Keyes of Columbia University for is valuable advice in te preparation of tis wor. Tis wor was supported by te United States Department of Energy under contract No. DE-AC- 9CH6. References [SL9] Strauss, H. R. and Longcope, D. W.: An Adaptive Finite Element Metod for Magnetoydrodynamics., Journal of Computational Pysics, 7:3 336, 99.