HIGH ACCURACY METHOD FOR MAGNETOHYDRODYNAMICS SYSTEM IN ELSÄSSER VARIABLES

Transcription

1 HIGH ACCURACY METHOD FOR MAGNETOHYDRODYNAMICS SYSTEM IN ELSÄSSER VARIABLES N. WILSON, A. LABOVSKY, C. TRENCHEA Key words. Magnetohydrodynaics, partitioned ethods, deferred correction, Elsässer variables. Abstract. A ethod has been developed recently by the third author, that allows for decoupling of the evolutionary full Magneto- HydroDynaics MHD syste in the Elsässer variables. The ethod entails the iplicit discretization of the subproble ters and the explicit discretization of coupling ters, and was proven to be unconditionally stable. In this paper we build on that result by introducing a high-order accurate deferred correction ethod, which also decouples the MHD syste. We perfor the full nuerical analysis of the ethod, proving the unconditional stability and second order accuracy of the two-step ethod. We also use a test proble to verify nuerically the claied convergence rate.. Introduction. The equations of agnetohydrodynaics MHD describe the otion of electrically conducting, incopressible flows in the presence of a agnetic field. When an electrically conducting fluid oves in a agnetic field, the agnetic field exerts forces which ay substantially odify the flow. Conversely, the flow itself gives rise to a second, induced field and thus odifies the agnetic field. Initiated by Alfvèn in 9 [], MHD is widely exploited in nuerous branches of science including astrophysics and geophysics [, 35, 6,,, 3, 6, 5], as well as engineering, e.g., liquid etal cooling of nuclear reactors [, 3, 38], process etallurgy [8], sea water propulsion [3]. The MHD flows entails two distinct physical processes: the otion of fluid is governed by hydrodynaics equations and the agnetic field is governed by Maxwell equations. One approach to solve the coupled proble is by onolithic ethods, or iplicit fully coupled algoriths, that are robust and stable, but quite deanding in coputational tie and resources. In these ethods, the globally coupled proble is assebled at each tie step and then solved iteratively. Partitioned ethods, which solve the coupled proble by successively solving the sub-physics probles [30], are another attractive and proising approach for solving MHD syste. Most terrestrial applications, in particular ost industrial and laboratory flows, involve sall agnetic Reynolds nuber. In this cases, while the agnetic field considerably alters the fluid otion, the induced field is usually found to be negligible by coparison with the iposed field [8, 36, 8]. Neglecting the induced agnetic field one can reduce the MHD systes to the significantly sipler Reduced MHD RMHD, for which several iplicit-explicit IMEX schees were studied in [9]. In this report we ai to iprove the accuracy of the first order ethod introduced in []. The ethod that we ai to develop for the evolutionary full MHD equations, at high agnetic Reynolds nuber in the Elsässer variables, also needs to be stable and allow for explicit-iplicit ipleentations with different tie scales. To that end, we eploy the spectral deferred correction SDC ethod, proposed for stiff ODEs by Dutt et al., [3], and further developed by Minion et al.; see [33, 3, 7] and the references therein. SDC ethods were studied and copared to intrinsically high-order ethods such as additive Runge-Kutta ethods and linear ultistep ethods based on BDFs, with the conclusion that the SDC ethods are at least coparable to the latter. In addition, achieving high accuracy for the turbulent NSE using Runge-Kutta-based ethods is very expensive, and the BDF-based ethods typically do not perfor well in probles where relevant tie scales associated with different ters in the equation are widely different; see, e.g., [7] for an exaple of an advection-diffusion-reaction proble for which the SDC is the best choice for high-accuracy teporal discretization. Departent of Matheatical Sciences, Michigan Technological University, Houghton, MI 993, USA, Eail: newilson@tu.edu and aelabovs@tu.edu. Departent of Matheatics, University of Pittsburgh, Pittsburgh, PA 560, USA, Eail: trenchea@pitt.edu. Partially supported by Air Force grant FA

2 The equations of agnetohydrodynaics describing the otion of an incopressible fluid flow in presence of a agnetic field are the following see, e.g. [8, 5, ] u + u u B B ν u + p = f, t u = 0, B t + u B B u ν B = g, B = 0, in Ω 0,T, where Ω is the fluid doain, u = u x,t,u x,t,u 3 x,t is the fluid velocity, px,t is the pressure, B = B x,t,b x,t,b 3 x,t is the agnetic field, f and g are external forces, ν is the kineatic viscosity and ν is the agnetic resistivity. The total agnetic field can be split in two parts B = B + b ean and fluctuations. We prescribe hoogeneous Dirichlet boundary conditions for u, and B = B on the boundary see [8] for typical agnetic boundary conditions. Then the Elsässer fields [] z + = u + b, z = u b,. erging the physical properties of the Navier-Stokes and Maxwell equations, suggest stable tie-splitting schees for the full MHD equations. The oentu equations, in the Elsässer variables, are z ± t B z ± + z z ± ν+ν z ± ν ν z + p = f ±,. while the continuity equations are z ± = 0. We note that the nonlinear interactions occur between the Alfvènic fluctuations z ±. The ean agnetic field plays an iportant role in MHD turbulence, for exaple it can ake the turbulence anisotropic; suppress the turbulence by decreasing energy cascade, etc. In the presence of a strong ean agnetic field, z + and z wavepackets travel in opposite directions with the phase velocity of B, and interact weakly. For Kologorov s and Iroshnikov/Kraichnan s phenoenological theories of MHD isotropic and anisotropic turbulence, see [5, 7, 9, 3,, 37, 7, 0, 3]. In a classical understanding, the deferred correction approach to solving ODEs is based on replacing the original ODE in our case, the syste of ODEs obtained fro the original PDEs by the Method of Lines with the corresponding Picard integral equation, discretizing the tie interval, solving the integral equation approxiately and then correcting the solution by solving a sequence of error equations on the sae grid with the sae schee; see [3] and [33] for the detailed atheatical presentation of SDC. In particular, the two-step Deferred Correction ethod introduced in this paper, perfors as follows. The first approxiation to the sought quantities in this case, the Elsässer variables z +,z is obtained by the stable and coputationally attractive first order accurate IMEX ethod of []. Then the second order accurate approxiation is coputed, which iproves the accuracy without sacrificing stability. Note that the second step utilizes the sae IMEX tie discretization as in the first step; only the right-hand side is odified by a known quantity, i.e, a known solution fro the first step. This results in the coputational attractiveness of the ethod: coputing two low-order accurate approxiations is uch less costly especially for very stiff probles than coputing a single higher-order approxiation.. Notation and Preliinaries. We consider a doain Ω R d d= or 3 to be a convex polygon or polyhedra. We denote the failiar Lesbegue easure spaces by L p Ω, and denote the L Ω inner product and induced nor by, and respectively. Additionally, we denote the L Ω nor by, and the nor associated with the Sobolev spaces W,k Ω H k Ω by k. All other nors will be clearly labeled. Throughout the article, we will ake use of the inequalities presented in the following lea. In Lea. and subsequent analysis we denote constants that are independent of ν and ν by C. The generic constant varies throughout this work. LEMMA.. If u,v,w H Ω, and u = 0 then u v,w = u w,v,

3 u v,v = 0, u v,w C u v w provided v L, u v,w C u v w provided u L, u v,w C u u v w, and u v,w C u u 3 v w w 3. These are classical results used in the study of Navier-Stokes equations and agnetohydrodynaics see for exaple [39, 0, 0]. In addition to the above results we will also eploy the following discrete Gronwall s lea, which is proved in [], in the error analysis below. LEMMA.. Let, H, and a n,b n,c n,d n be nonnegative nubers n = 0,...,N satisfying Then for all > 0, a N + a N + N N N b n b n exp d n a n + N dn N N c n + H. c n + H. Note that this version of Gronwall s lea places no restriction on the size of the tiestep. 3. The iplicit-explicit partitioned schees. The heart of any partitioned ethod, aiing at decoupling the two physically interconnected subprobles, is its treatent of the coupling ters. The ethod we study herein has the coupling ters lagged or extrapolated in a careful way that preserves stability. 3.. First order unconditionally stable IMEX partitioned schee. The ethod proposed and analyzed in [] has the coupling ters lagged, thus the syste uncouples into two subproble solves. It approxiates the oentu equations. and continuity equations in the Elsässer variables by the following first-order IMEX schee backward-euler forward-euler 0z ± n+ 0 z ± n B 0 z ± n+ + 0z n 0 z ± n+ 3. ν + ν 0 z ± n+ ν ν 0 z n + 0 p ± n+ = f ± t n+, z ± n+ = The schee is odular, i.e., the variables 0 z + and 0 z are decoupled, and is unconditionally absolute-stable. We note that the pre-subscript occurring on the variables 0 z ± n is used to denote the first order IMEX approxiation to the Elsässer variables zt n ± respectively. As entioned before the unconditional stability of the IMEX ethod is proven in []. Thus, we restrict our attention to showing the ethod is first order accurate in tie. We note the ter η n occurs regularly in the analysis below. The value of η n is between the tiesteps t n and t n+ the arises fro our use of Taylor series. So, it is unknown and varies in each occurrence. LEMMA 3.. Given a final tie T > 0 and tiestep > 0 let t n = n for n = 0,,...,N. Let 0 z ± n denote the IMEX approxiation of z ± t n, and let 0e ± n := z ± t n 0 z ± n. Then provided the true solution satisfies the regularity assuptions z ± L 0,T ;L Ω, t z ± L 0,T ;L Ω, tt z ± L 0,T ;L Ω, 3

4 t z ± L 0,T ;L Ω, the following error estiate holds 0 e + N + 0 e νν N + ν + ν C ν + ν exp νν N N N 0 e + n + 0 e n z t n + z + t n tt z + η n + t z η n z + t n + ν ν t z η n + t z + η n + tt z η n + t z + η n z t n. 3.3 Proof. We begin by rewriting the first continuous oentu equation as z+ t n+ z + t n + z t n z + t n+ B z + t n+ ν + ν z + t n+ ν ν z t n = z+ t n+ z + t n t z + t n+ pt n+ + z t n z + t n+ z t n+ z + t n+ + ν ν z t n+ ν ν z t n + f + t n+. The first oentu equation for the IMEX ethod is 0z + n+ 0 z + n + 0 z n 0 z + n+ B 0 z + n+ ν + ν 0 z + n+ 3. ν ν 0 z n + 0 p + n+ = f + t n The second continuous oentu equation ay be expressed as z t n+ z t n + z + t n z t n+ + B z t n+ ν + ν z t n+ ν ν z + t n = z t n+ z t n t z t n+ pt n+ + z + t n z t n+ z + t n+ z t n+ + ν ν The second oentu equation for the IMEX ethod satisfies 0z n+ 0 z n + 0 z + n 0 z n+ + B 0 z n+ ν + ν z t n+ ν ν z + t n + f t n z n+ ν ν 0 z + n + 0 p n+ = f t n+. 3.7

5 Subtracting 3.5 fro 3. and ultiplying by 0 e + n+, subtracting 3.7 fro 3.6 and ultiplying by 0e n+, adding the resulting equations and reducing gives 0e + n+ 0 e + n + 0e n+ 0 e n + ν ν ν + ν 0e + n+ 0 e + n + 0 e n+ 0 e n + νν ν + ν 0 e + n+ + 0 e n+ 0 z n 0 z + n+, 0e + n+ z t n z + t n+, 0 e + n+ + z + t n z t n+, 0 e n+ z+ t n+ z t n+, 0 e n+ + 0 z + n 0 z n+, 0e n+ z+ t n z t n+, 0 e n+ + z t n z + t n+, 0 e + n+ z t n+ z + t n+, 0 e + n+ + z t n+ z t n + z+ t n+ z + t n + ν ν + ν ν t z + t n+, 0 e n+ t z + t n+, 0 e + n+ z t n+ z t n, 0 e + n+ z + t n+ z + t n, 0 e n+. The proof continues by bounding the ters occurring on the RHS of 3.8 as follows 0 z n 0 z + n+, 0e + n+ z t n z + t n+, 0 e + n+ = 0 e + n z + t n+, 0 e + n+ 3.8 Cγ 0 e + n z + t n+ + γ 0 e + n+, 3.9 z + t n z t n+, 0 e n+ z+ t n+ z t n+, 0 e n+ = z + t n z + t n+ z t n+, 0 e n+ C γ t z + η n z t n+ + γ 0 e n z + n 0 z n+, 0e n+ z+ t n z t n+, 0 e n+ = 0 e + n z t n+, 0 e n+ Cγ 0 e + n z t n+ + γ 0 e n+ 3. z t n z + t n+, 0 e + n+ z t n+ z + t n+, 0 e + n+ = z t n z t n+ z + t n+, 0 e + n+ Cγ t z η n z + t n + γ 0 e + n+ 3. z t n+ z t n t z + t n+, 0 e n+ C γ tt z η n + γ 0 e n+ 3.3 z+ t n+ z + t n t z + t n+, 0 e + n+ C γ tt z + η n + γ 0 e + n+ 3. 5

6 Specifying γ = ν ν z t n+ z t n, 0 e + n+ C ν ν γ t z η n + γ 0 e + n+ 3.5 ν ν z + t n+ z + t n, 0 e n+ C ν ν γ t z + η n + γ 0 e n+ 3.6 νν 8ν+ν, substituting into 3.8, and rearranging gives 0e + n+ 0 e + n + 0e n+ 0 e n + ν ν ν + ν 0e + n+ 0 e + n + 0 e n+ 0 e n + νν ν + ν 0e + n+ + 0 e n+ C ν + ν t z + η n z t n+ + t z η n z + t n νν + tt z η n + tt z + η n + ν ν t z η n + ν ν t z + η n +C ν + ν νν 0 e +n + 0 e +n z + t n+ + z t n The proof is finished by ultiplying 3.7 by, suing fro n = 0 to N, dropping nonnegative LHS ters, and applying Gronwall s inequality. 3.. Second order unconditionally stable SISDC partitioned schee. Having shown the IMEX ethod is linear, unconditionally stable, odular and first order accurate in tie we seek to develop a ore accurate ethod that retains the good qualities of the IMEX ethod. To this end we eploy the spectral deferred correction technique for further details of this technique see [3, 33, ]. The second order sei-iplicit spectral deferred correction ethod is as follows: after coputing first order approxiations, 0 z ± n, 0 p ± n and 0 z ± n+, 0 p ± n+ using for exaple the IMEX ethod above of. at tie t n and t n+ respectively we seek to copute z ± n+, p ± n+ satisfying z ± n+ z ± n ν + ν z ± n+ ν ν z ± n B z ± n+ + z n z ± n+ + p ± n+ = B 0 z ± n+ ν + ν ± B 0 z ± n+ ± B 0 z ± n + ν + ν + ν ν 0 z ± n+ + ν + ν 0 z ± n 0 z ± n+ ν ν 0 z n 0 z ± n+ + ν ν 0 z ± n 0z n+ 0z ± n+ 0z n+ 0z ± n 0 p ± n+ + 0p ± n+ + 0p ± n + f t n+ ± + f t n ±, z ± n+ = 0. 6

7 LEMMA 3.. Given a final tie T > 0 and tiestep > 0, let 0 z ± n denote the IMEX solution at tie t n = n for n =,,...,N. Then the solutions to the SISDC ethod are unconditionally stable and satisfy N z + N + z νν N + k= ν + ν z + k + z k + ν ν z +N ν + ν + z N C ν + ν νν N k=0 B + ν + ν + ν ν 0 z + k+ + 0 z + k + 0 z k+ + 0 z k + 0 z + k+ + 0 z + k 0 z k+ + 0 z k + f + t k+ + f + t k + f t k+ + f t k + zt zt 0 + ν ν ν + ν zt zt 0. Proof. Multiplying the SISDC oentu equations by z + n+ and z n+ respectively, applying continuity equations and polarization identity, and adding the relations gives z + n+ z + n + z + n+ z+ n + z n+ z n + z n+ z n = B 0 z + n+, z + n+ + ν + ν + ν + ν + ν + ν z + n+ + ν ν z n, z + n+ z n+ + ν ν z + n, z n+ 0 z + n+, z + n+ + ν ν 0 z n, z + n+ + B 0 z + n+, z + n+ + B 0 z + n, z + n+ ν + ν 0 z n, z + n+ ν ν 0 z n+, z + n+ + 0z n 0 z + n+, z + n+ 0z n+ 0z + n+, z + n+ 0z n 0 z + n, z + n+ + f + t n+, z + n+ + f + t n, z + n B 0 z n+, z n+ + ν + ν 0 z n+, z n+ + ν ν 0 z + n, z n+ B 0 z n+, z n+ B 0 z n, z n+ ν + ν 0 z n, z n+ ν ν 0 z n+, z n+ + 0z + n 0 z n+, z n+ 0z + n+ 0z n+, z n+ 0z + n 0 z n, z n+ + f t n+, z + n+ + f t n, z + n+. Lower bounding dissipation ters on the LHS of 3.8 using the Cauchy-Schwarz inequality and polarization 7

8 identity for further details see [] and dropping nonnegative ters gives z n+ z + n + z n+ z n + νν ν + ν z + n+ + z n+ + ν ν z + n+ ν + ν z + n + z n+ z n B 0 z + n+, z + n+ + ν + ν 0 z + n+, z + n+ + ν ν 0 z n, z + n+ + B 0 z + n+, z + n+ + B 0 z + n, z + n+ ν + ν 0 z n, z + n+ ν ν 0 z n+, z + n+ + 0z n 0 z + n+, z + n+ 0z n+ 0z + n+, z + n+ 0z n 0 z + n, z + n+ + f + t n+, z + n f + t n, z + n+ + B 0 z n+, z n+ + ν + ν 0 z n+, z n+ + ν ν 0 z + n, z n+ B 0 z n+, z n+ B 0 z n, z n+ ν + ν 0 z n, z n+ ν ν 0 z n+, z n+ + 0z + n 0 z n+, z n+ 0z + n+ 0z n+, z n+ 0z + n 0 z n, z n+ + f t n+, z + n+ + f t n, z + n+. Majorizing the RHS ters of 3.9 with standard inequalities gives z n+ z + n + z n+ z n + νν ν + ν z + n+ + z n+ + ν ν z + n+ ν + ν z + n + z n+ z n C ν + ν ν ν B + ν + ν + ν ν 0 z + n+ + 0 z + n + 0 z n+ + 0 z n + 0 z + n+ + 0 z + n 0 z n+ + 0 z n + f + n+ + f + n + f n+ + f n. 3.0 Multiplying by and suing over tiesteps yields the desired result. The purpose of adding the correction step is to develop a ore accurate nuerical ethod and so we seek to show that the SISDC solutions are in fact second order accurate. To this end we state and prove the following lea, which is necessary for the proof of the accuracy of the SISDC ethod. 8

9 LEMMA 3.3. Given a final tie T > 0 and tiestep > 0 let 0 z ± n be the IMEX approxiation to zn ± for n =,,...,N. Let 0 e ± n = zt n ± 0 z ± n Provided the true solution satisfies the additional regularity assuptions ttt z ± L 0,T ;L Ω, t z ± L 0,T ;L Ω, t z ± L 0,T ;L Ω the discrete tie derivative of the IMEX is first order accurate in tie and satisfies 0 e + N 0e + N + 0 e N 0e N N νν 0 e + ν + ν + N 0e + N + 0 e N 0e N C ν + ν N exp νν z + t n + z t n+ N ttt z + η n + ttt z η n + t z + η n + t z η n t z + η n + tt z η n z + t n 3. + tt z + η n t z t n + t z η n 0 e + n+ + t z + η n 0 e n + t z + η n 0 e n+ + t z η n 0 e + n + 0 e + n + 0 e n Proof. For ease of notation we begin by defining s n+ ± := 0e ± n+ 0e ± n. Next consider 3. and 3.5 at tiesteps n + and n. Subtracting 3.5 fro 3. at tiestep n + gives an equation involving 0 e + n+, and subtracting 3.5 fro 3. at tiestp n gives an equation involving 0 e + n. We then subtract the new equations involving the IMEX errors to get an equation that involves s + n+. We siilarly derive an equation involving 9

10 s n+. Multiplying the respective equations by s+ n+ and s n+, adding the equations, and reducing gives s+ n+ s + n + s n+ s n + ν + ν s + n+ + s n+ + ν ν s + n+ ν + ν s + n + s n+ s n = 0z n 0 z + n+,s+ n+ z t n z + t n+,s + n+ + z t n z + t n,s + n+ 0z n 0z + n,s + n+ + 0z + n 0 z n+,s n+ z+ t n z t n+,s n+ + z+ t n z t n,s n+ 0z + n 0z n,s n+ + z t n z + t n+,s + n+ z t n+ z + t n+,s + n+ + z t n z + t n,s + n+ z t n z + t n,s + n z+ t n z t n+,s n+ z+ t n+ z t n+,s n+ + z+ t n z t n,s n+ z+ t n z t n,s n+ z + t n+ z + t n + t z + t n+,s + n+ z + t n z + t n z t n+ z t n t z + t n,s + n+ + t z t n+,s n+ z t n z t n t z t n,s n+. The nonlinear ters of 3. have been separated into four groups. Treatent of the first group of nonlinear begins by applying the identity ab cd = ab d + a cd twice which yields 0z n 0 z + n+,s+ n+ z t n z + t n+,s + n+ + z t n z + t n,s + n+ 0z n 0z + n,s + n+ = 0z n 0 e + n+,s+ n+ 0e n z + t n+,s + n z t n 0 e + n,s + n+ + 0e n 0z + n,s + n+. 0

11 Adding two zero ters to to 3.3 regrouping and applying the sae identity gives 0z n 0 z + n+,s+ n+ z t n z + t n+,s + n+ + z t n z + t n,s + n+ 0z n 0z + n,s + n+ = z t n 0 e + n+,s+ n+ 0z n 0 e + n+,s+ n+ ± zt n 0 e + n+,s+ n+ + 0e n 0z + n,s + n+ 0e n z+ t n+,s + n+ 3. ± 0e n z+ t n,s + n+ = z t n z t n 0 e + n+,s+ n+ + 0s n 0 e + n+,s+ n+ + 0e n z+ t n z + t n+,s + n+ s n z + t n+,s + n+ We now bound the ters in 3. with standard inequalities as follows z t n z t n 0 e + n+,s+ n+ C t z η n 0 e + n+ s+ n+ Cγ t z η n 0 e + n+ + γ s + n+, 0 s n 0 e + n+,s+ n+ C 0 s n 0 s n 0 e + n+ s+ n+ Cγ C 0 s n 0 s n 0 e + n+ + γ s + n+ Cγ 0 s n +Cγ 0 s n 0 e + n+ + γ s + n+, 0e n z+ t n z + t n+,s + n+ C 0 e n tz + η n s + n+ Cγ 0 e n t z + η n + γ s + n+, s n z + t n+,s + n+ s n z + t n+ s + n+ Cγ s n z + t n+ + γ s + n Cobining gives the following bound for the first group of nonlinear ters 0z n 0 z + n+,s+ n+ z t n z + t n+,s + n+ + z t n z + t n,s + n+ 0z n 0z + n,s + n+ Cγ t z η n 0 e + n+ + 0 s n + s n 0 e + n+ 0 e n t z + η n + s n z + t n+ + γ s + n+. 3.9

12 Siilar treatent of the second group of nonlinear ters yields the following bound 0z + n 0 z n+,s n+ z+ t n z t n+,s n+ + z+ t n z t n,s n+ 0z + n 0z n,s n+ Cγ t z + η n 0 e n+ + 0 s + n + s + n 0 e n+ + 0 e + n t z η n + s + n z t n+ + γ s n We now derive bounds for the third group of nonlinear ters. Grouping the ters linearly, applying the identity ab-cd = ab-d+a-cd, using Taylor-series expansions, and standard bounds on the nonlinearity gives z t n z + t n+,s + n+ z t n+ z + t n+,s + n+ + z t n z + t n,s + n+ z t n z + t n,s + n+ = z t n z t n+ z + t n+,s + n+ + z t n z t n z + t n,s + n+ = z t n z t n+ z + t n+ z + t n,s + n+ + z t n z t n+ z + t n,s + n+ Cγ t z η n t z + η n +Cγ tt z η n z + t n + γ s + n Siilar treatent of the fourth group of nonlinear ters gives z+ t n z t n+,s n+ z+ t n+ z t n+,s n+ + z+ t n z t n,s n+ z+ t n z t n,s n+ Cγ t z + η n t z η n +Cγ tt z + η n z t n + γ s n+. Applying Taylor series to the reaining linear ters of 3. yields z + t n+ z + t n t z + t n+,s + n+ z + t n z + t n t z + t n,s + n+ Cγ ttt z + η n + γ s + n

13 z t n+ z t n t z t n+,s n+ z t n z t n t z t n,s n+ Cγ ttt z η n + γ s n Having bounded all RHS ters of 3. we continue by choosing γ = ν+ν, substituting 3.9, 3.30, 3.3, 3.3, 3.33, and 3.3 into 3., and rearranging. Multiplying the resulting equation by gives νν s + n+ s + n + s n+ s n νν + ν + ν s+ n+ + s n+ νν + ν + ν s+ n+ s + n + s n+ s n C ν + ν s +n + s n z + t n+ + z t n+ νν +C ν + ν t z η n νν 0 e + n+ + 0 e n t z + η n + t z + η n 0 e n+ + 0 e + n t z η n + t z η n t z + η n + tt z η n z + t n + t z + η n t z η n + tt z + η n z t n + ttt z + η n + ttt z η n C ν + ν s + n 0 e n+ νν +C ν + ν s n 0 e + n+ νν. To finish deriving the error bound requires an application of Gronwall s inequality. However, there are two subtleties that prevent us fro doing this iediately. The first coplication is the last two ters of 3.35 involve s ± n, and so they require further treatent. Recall that 0 e ± n denotes the error in the IMEX solution, and so we have the following 0 e + n+ = 0e ± n+ C Using 3.36 we derive bounds for the last two ters of 3.35 as follows s ± n 0 e n+ C 3 s ± n C e ± n +C e ± n The other subtlety that requires our attention is it is only possible to su fro 3.35 n = to N, because s ± 0 = 0e ± 0 0 e ± is not defined, and this leaves nonpositive ters on the LHS that ust be dealt with. After we su 3.35 fro n = to N we are left with the ters s +, s +, s, and s on the LHS. To apply Gronwall s inequality requires oving these ters to the RHS, and to yield the desired result we need these bounded by a ultiple of. Recall that the IMEX ethod is a first order accurate ethod, which iplies the local error 0 e ± in the ethod is second order accurate. Thus, we have the following bound s + = 0e + 0e + 0 = 0 e + C Bounds for s +, s, and s can be derived siilarly. 3

14 Substituting 3.37 in to 3.35, suing fro n = to N, rearranging and applying the discrete Gronwall lea finishes the proof. We now state and prove the ain result, that solutions found with the SISDC ethod are second order accurate. THEOREM 3.. Given final tie T > 0 and tiestep > 0 let 0 z ± n and z ± n respectively denote the IMEX and SISDC approxiations to z ± at tie t n = n for n =,,...,N, and let e ± n := z ± t n z ± n. Additionally, we let s ± n+ = 0e ± n+ 0e ± n for n =,,...,N. Provided the true solution satisfies the additional regularity z ± L 0,T ;L, tt z z ± L 0,T ;L, tt z ± L 0,T ;L, and tt f +, tt f L 0,T ;L then the SISDC approxiation is second order accurate in tie and satisfies e + N + e νν N + ν + ν N C ν + ν exp νν N e + n + e n z + t n + z t n N C ν + ν { B s + n+ νν + B s n+ + z t n s + n+ + 0 e n s + n+ + 0 z n s + n+ + s n+ 0 e + n+ + t z η n 0 e + n+ + s n+ z + t n+ + 0 e n+ t z + η n + s n+ z + t n z + t n s n+ + 0 e + n s n+ + 0 z + n s n+ + s + n+ 0 e n+ + t z + η n 0 e n+ + s + n+ z t n+ + 0 e + n+ t z η n + s + n+ z t n + ν + ν s + n+ + ν ν s n+ + ν + ν s n+ + ν ν s + n+ + tt Fη n + tt Fη n }. Proof. We begin the proof by defining F + and F as follows F + := B z + z z + + ν + ν z + + ν ν z + f + p, 3.0 F := B z + z z + + ν + ν z + + ν ν z + f p. 3. We ay express the first continuous oentu equation as z+ t n+ z + t n = tn+ t n F + dτ.

15 Applying the trapezoid rule to the integral gives z+ t n+ z + t n ν + ν z + t n+ ν ν z t n = B z + t n+ + B z + t n z t n+ z + t n+ z t n z + t n ν + ν + ν ν z + t n+ + ν + ν z + t n z t n+ ν ν z t n + f + t n+ + f + t n pt n+ pt n +C tt F + η n. 3. The first oentu equation for the SISDC ethod is z + n+ z + n ν + ν z + n+ ν ν z + n = B z + n+ B 0 z + n+ ν + ν + B 0 z + n+ + B 0 z + n + ν + ν + ν ν 0 z + n+ ν ν 0 z n 0 z + n+ + ν + ν 0 z + n 0 z n+ + ν ν 0 z n z n z + n+ + 0z n 0 z + n+ 0z n+ 0z + n+ 0z n 0 z + n + f + t n+ + f + t n. 3.3 We ay siilarly express the second oentu equation as z t n+ z t n = tn+ F dτ. t n Applying the trapezoid rule to the integral gives z t n+ z t n ν + ν z t n+ ν ν z + t n = B z + t n+ B z + t n z+ t n+ z t n+ z+ t n z t n ν + ν + ν ν z t n+ + ν + ν z t n z + t n+ ν ν z + t n + f t n+ + f t n pt n+ pt n +C tt F η n. 3. The second oentu equation for the SISDC ethod is z n+ z n ν + ν z n+ ν ν z n = B z + n+ + B 0 z + n+ ν + ν B 0 z + n+ B 0 z + n + ν + ν + ν ν 0 z + n+ ν ν 0 z n 0 z + n+ + ν + ν 0 z + n 0 z n+ + ν ν 0 z n z + n z n+ + 0z + n 0 z n+ 5

16 0z + n+ 0z n+ 0z + n 0 z n + f t n+ + f t n. 3.5 Subtracting 3.3 fro 3. and ultiplying the result by e + n+ gives an equation in ters of e + n+. Siilarly, subtracting 3.5 fro 3. and ultiplying the result by e n+ gives an equation in ters of e n+. Adding these relations, reducing, and lower bounding dissipation ters gives e + n+ e + n + e n+ e n + νν ν + ν e + n+ + e n+ + ν ν ν + ν e + n+ e + n + e n+ e n B z + t n+, e + n+ + B z + t n, e + n+ B z + n+, e + n+ + B 0 z + n+, e + n+ B 0 z + n, e + n+ + B z t n+, e n+ B z t n, e n+ + B z n+, e n+ B 0 z n+, e n+ + z t n+ z + t n+, e + n+ z t n z + t n, e + n+ + z n z + n+, e + n+ 0 z n 0 z + n+, e + n+ + 0z n+ 0z + n+, e + n+ + 0z n 0 z + n, e + n+ + z+ t n+ z t n+, e n+ z+ t n z t n, e n+ + z + n z n+, e n+ 0 z + n 0 z n+, e n+ + 0z + n+ 0z n+, e n+ + ν + 0z + n 0 z n, e n+ ν + z + t n+,, e + n+ ν + ν ν ν ν ν z + t n,, e + n+ + ν ν z t n,, e + n+ z t n+,, e + n+ ν + ν 0 z + n+,, e + n+ 0 z n,, e + n+ + ν ν 0 z n+,, e + n+ ν + ν z t n+,, e n+ + ν + ν 0 z n,, e + n+ + ν + ν + ν ν ν ν z t n,, e n+ ν ν z + t n+,, e n+ z + t n,, e n+ ν + ν 0 z n+,, e n+ 0 z + n,, e n+ + ν + ν 0 z n,, e n+ 6

17 + ν ν 0 z n+,, e n+ + C tt F +, e + n+ +C tt F, e n The RHS ters of 3.6 are separated into seven groups. To derive the bound for the first group we add and subtract B z + t n+, e + n+ to see B z + t n+, e + n+ + B z + t n, e + n+ B z + n+, e + n+ + B 0 z + n+, e + n+ B 0 z + n, e + n+ = B z + t n+, e + n+ B z + n+, e + n+ this line is 0 + B 0 z + n+, e + n+ B z + t n+, e + n B z + t n, e + n+ B 0 z + n, e + n+ = B 0 e + n+ 0e + n, e + n+ Cγ B s + n+ + γ e + n+. The second group of ters in bounded as follows B z t n+, e n+ B z t n, e n+ + B z n+, e n+ B 0 z n+, e n+ + B 0 z n, e n+ 3.8 Cγ B s n+ + γ e n+. To bound the third group of ters in 3.6 we add and subtract the ter z t n z + t n+, e + n+ to get z n z + n+, e + n+ z t n z + t n+, e + n+ +z t n z + t n+, e + n+ 0z n 0 z + n+, e + n+ + 0z n+ 0z + n+, e + n+ z t n+ z + t n+, e + n+ + 0z n 0 z + n, e + n+ z t n z + t n, e + n Using the identity ab cd = ab d + a cd on the ters of 3.9 gives 0 e n z + t n+, e + n+ +z t n 0 e + n+, e + n+ + 0e n 0 z + n+, e + n+ 0z n+ 0e + n+, e + n+ 0e n+ z+ t n+, e + n+ 0z n 0 e + n, e + n+ 0e n z + t n, e + n We continue to derive the bound for the third group of nonlinear ters by focussing on the third through eighth ters in Cobining the third, fifth, and seventh ters in 3.50 and adding and subtracting 7

18 0z n 0 e + n, e + n+ gives z t n 0 e + n+, e + n+ 0z n 0 e + n, e + n+ + 0z n 0 e + n, e + n+ 0z n+ 0e + n+, e + n+ = z t n 0 e + n+ 0e + n, e + n+ + 0e n 0 e + n, e + n z n 0 e + n 0 e + n+, e + n+ + 0z n 0 z n+ 0e + n+, e + n+. The last ter in 3.5 requires further attention 0z n 0 z n+ 0e + n, e + n+ ± z t n z t n+, 0 e + n+, e + n+ = 0e n+ 0e n 0 e + n+, e + n z t n z t n+, 0 e + n+, e + n+. Next, cobining the fourth, sixth, and eighth ters in 3.50 and adding and subtracting 0e n+ 0z + t n+, e + n+ gives 0 e n 0 z + n+, e + n+ 0e n+ z+ t n+, e + n+ + 0e n+ 0z + t n+, e + n+ 0e n 0 z + t n, e + n+ = 0 e n 0 e + n+, e + n+ + 0e n 0 e n+ z+ t n+, e + n+ + 0e n+ z+ t n+ z + t n, e + n e n+ 0e n z + t n, e + n+. Substituting, gives the following equality z t n+ z + t n+, e + n+ z t n z + t n, e + n+ + z n z + n+, e + n+ 0z n 0 z + n+, e + n+ + 0z n+ 0z + n+, e + n+ + 0z n 0 z + n, e + n+ = e n z + t n+, e + n+ + z t n 0 e + n+ 0e + n, e + n+ + 0 e n 0 e + n 0 e + n+, e + n+ + 0z n 0 e + n 0 e + n+, e + n e n+ 0e n 0 e + n+, e + n+ + 0e n+ 0e n z + t n, e + n+ + 0 e n 0 e n+ z+ t n+, e + n+ + 0e n+ z+ t n+ z + t n, e + n+ + z t n z t n+, 0 e + n+, e + n+. 8

19 Applying standard inequalities gives the following upperbound for the third group of nonlinear ters in 3.6 z t n+ z + t n+, e + n+ z t n z + t n, e + n+ + z n z + n+, e + n+ 0z n 0 z + n+, e + n+ + 0z n+ 0z + n+, e + n+ + 0z n 0 z + n, e + n+ Cγ e n z + t n+ +Cγ z t n s + n e n s + n+ + 0 z n s + n+ + s n+ 0 e + n+ + t z η n 0 e + n+ + s n+ z + t n+ + 0 e n+ t z + η n + s n+ z + t n + 9γ e + n+. The bound for the fourth group of ters in 3.6 is derived in a siilar way. The bound is z+ t n+ z t n+, e n+ z+ t n z t n, e n+ + z + n z n+, e n+ 0z + n 0 z n+, e n+ + 0z + n+ 0z n+, e n+ + 0z + n 0 z n, e n+ Cγ e + n z t n+ +Cγ z + t n s n e + n s n+ + 0 z + n s n+ + s + n+ 0 e n+ + t z + η n 0 e n+ + s + n+ z t n+ + 0 e + n+ t z η n + s + n+ z t n + 9γ e n+. Cobing the ters in the fifth group of ters in 3.6 and applying standard inequalities gives ν + ν z + t n+,, e + n+ ν + ν 0 z + n+,, e + n+ + ν ν + ν + ν z t n,, e + n+ ν ν 0 z n,, e + n+ + ν ν 0 z n,, e + n+ ν + ν z + t n,, e + n+ 0 z n+,, e + n+ ν ν z t n+,, e + n+ C γ ν + ν s + n+ +C γ ν ν s n+ + γ e + n

20 The sixth group of ters in 3.6 are bounded siilarly ν + ν z t n+,, e n+ ν + ν 0 z n+,, e n+ + ν + ν + ν ν 0 z n,, e n+ ν + ν z t n,, e n+ + ν ν 0 z + n+,, e n+ ν ν z + t n+,, e n+ z + t n,, e n+ ν ν 0 z + n,, e n+ C γ ν + ν s n+ +C γ ν ν s + n+ + γ e n The reaining two ters in 3.6 are bounded as follows tt F +, e + n+ C γ tt F + η n + γ e + n+, 3.59 νν tt F, e n+ C γ tt F η n + γ e n Specifying γ = 6ν+ν, substituting the bounds 3.7, 3.8, 3.55, 3.56, 3.57, 3.58, 3.59, and 3.60 in to 3.6 and rearranging gives e + n+ e + n + e n+ e n + νν ν + ν e + n+ + e n+ + ν ν ν + ν e + n+ e + n + e n+ + e n C ν + ν B s + n+ νν + B s n+ + z t n s + n+ + 0 e n s + n+ + 0 z n s + n+ + s n+ 0 e + n+ + t z η n 0 e + n+ + s n+ z + t n+ + 0 e n+ t z + η n s n+ z + t n + z + t n s n+ + 0 e + n s n+ + 0 z + n s n+ + s + n+ 0 e n+ + t z + η n 0 e n+ + s + n+ z t n+ + 0 e + n+ t z η n + s + n+ z t n + ν + ν s + n+ + ν ν s n+ + ν + ν s n+ + ν ν s + n+ + tt F + η n + tt F η n +C ν + ν e n + e + n z + t n+ + z t n+ νν. Multiplying 3.6 by, suing fro n = 0 to N, and applying Gronwall s inequality finishes the proof.. Coputational results. Consider a well-known test proble for the -D NSE: two-diensional wave propagation considered on a square [0.5,.5] [0.5,.5]. This exaple is chosen because the solutions are 0

21 varying soothly in space so that it is easier to track the error due to the teporal discretization; for ore details on the traveling wave test proble see, e.g., [3, 6, 9] and the references therein. Let the flow be electrically conducting, and introduce the tie-varying agnetic field so that the true solution in Elsässer variables z +,z is z cosπx tsinπy te = 8π tre + 0.y + 3 e tre 0.5sinπx tcosπy te 8π tre + 0.x + 3, e tre z cosπx tsinπy te = 8π tre 0.y + 3 e tre 0.5sinπx tcosπy te 8π tre 0.x + 3, e tre p = 6 cosπx t + cosπy te 6π tre. We test the case of a lainar flow, and we also want Re Re to avoid unnecessary cancelation of ters, therefore we choose Re =,Re = 0. Galerkin finite eleent ethod is eployed, using the Taylor-Hood eleents piecewise quadratic polynoials for z + and z and piecewise linear polynoials for p. The results presented were obtained by using the software package FreeFEM + +. We copare the true solution to a solution obtained by our two-step deferred correction ethod. It follows fro the theoretical results that an error of the order Ok + h is to be expected when approxiating the true solution z +,z by the first-step variables w,z this is the IMEX ethod. Then, the correction-step variables cw,cz should approxiate the true solution z +,z with Ok + h. We set the tie step equal to the esh size, k = h, to verify the claied second-order accuracy of the ethod. The tables below deonstrate the first-order accuracy of the IMEX approxiation w, z and the second-order accuracy of the correction step approxiation cw, cz. The error is easured in the spatial nor L [0.5,.5] at the final tie level T =, N = T = h. TABLE. First-order approxiation, IMEX, Re =, Re = 0, T =. h z + T w N L Ω rate z T z N L Ω rate / / / / / TABLE. Correction step approxiation second order, Re =, Re = 0, T =. h z + T cw N L Ω rate z T cz N L Ω rate / / / / / The convergence rates in Tables.,. were coputed using the step sizes h = /,/6,/6, and the correction step approxiation is clearly of the second order; if the data fro h = /3 and h = /6 is used, then it follows fro Table. that the corresponding IMEX convergence rates are.07 and.09. Hence, the coputational results are consistent with the claied accuracy of the ethod.

22 5. Conclusions. When solving full evolutionary MHD systes, it is usually ore coputationally cheap to eploy partitioned rather than onolithic ethods. These ethods ai at decoupling the MHD syste by successively solving the two sub-physics probles. Not only this approach is coputationally attractive for large N it is uch cheaper to solve the N N subsyste twice, than the full N N one tie, but it also allows for parallelization and the use of legacy codes for the physical subprobles. An unconditionally stable although only first-order accurate IMEX ethod was proposed in [], which decouples the full MHD syste using the explicit discretization of the coupling ters. In this paper, we have introduced and thoroughly studied the higher-order accurate ethod, which utilizes the deferred correction approach, built on the foreentioned IMEX schee. The choice of the deferred correction as opposed to other high order ethods like Adaptive Runge-Kutta or the BDFs is based on the fact that different ters in the MHD systes can evolve on different tie scales - and the deferred correction is known to be well tailored for such probles. We proved the unconditional stability of our ethod and for the case of the two-step ethod its second order accuracy. The claied accuracy was then nuerically verified on a test proble of the wave traveling in the presence of the agnetic field. This test proble was chosen because the error is affected ainly by its teporal coponent, and not the spatial coponent the study of possibly different spatial discretizations is outside of the scope of this paper. As a result, we obtained a ethod for solving full evolutionary MHD systes in Elsässer variables, which is fast, unconditionally stable and second order accurate, and allows for the usage of different sizes of tie steps for different ters of the MHD syste. REFERENCES [] H. ALFVÉN, Existence of electroagnetic-hydrodynaic waves, Nature, 50 9, p. 05. [] L. BARLEON, V. CASAL, AND L. LENHART, MHD flow in liquid-etal-cooled blankets, Fusion Engineering and Design, 99, pp. 0. [3] J. D. BARROW, R. MAARTENS, AND C. G. TSAGAS, Cosology with inhoogeneous agnetic fields, Phys. Rep., 9 007, pp [] D. BISKAMP, Nonlinear agnetohydrodynaics, vol. of Cabridge Monographs on Plasa Physics, Cabridge University Press, Cabridge, 993. [5] D. BISKAMP, Magnetohydrodynaic turbulence, Cabridge: Cabridge University Press, 003. [6] P. BODENHEIMER, G. P. LAUGHLIN, M. RÓŻYCZKA, AND H. W. YORKE, Nuerical ethods in astrophysics, Series in Astronoy and Astrophysics, Taylor & Francis, New York, 007. [7] A. BOURLIOUX, A. T. LAYTON, AND M. L. MINION, High-order ulti-iplicit spectral deferred correction ethods for probles of reactive flow, J. Coput. Phys., , pp [8] P. A. DAVIDSON, An introduction to agnetohydrodynaics, Cabridge Texts in Applied Matheatics, Cabridge University Press, Cabridge, 00. [9] M. DOBROWOLNY, A. MANGENEY, AND P. VELTRI, Fully developed anisotropic hydroagnetic turbulence in interplanetary space, Phys. Rev. Lett., 5 980, pp. 7. [0] C. R. DOERING AND J. D. GIBBON, Applied analysis of the Navier-Stokes equations, Cabridge Texts in Applied Matheatics, Cabridge University Press, Cabridge, 995. [] E. DORMY AND M. NÚÑEZ, Introduction [Special issue: Magnetohydrodynaics in astrophysics and geophysics], Geophys. Astrophys. Fluid Dyn., 0 007, p. 69. [] E. DORMY AND A. M. SOWARD, eds., Matheatical aspects of natural dynaos, vol. 3 of Fluid Mechanics of Astrophysics and Geophysics, Grenoble Sciences. Universite Joseph Fourier, Grenoble, 007. [3] A. DUTT, L. GREENGARD, AND V. ROKHLIN, Spectral deferred correction ethods for ordinary differential equations, BIT, 0 000, pp. 66. [] W. ELSÄSSER, The hydroagnetic equations, Phys. Rev., II. Ser., , p. 83. [5] A. FIERROS PALACIOS, The Hailton-type principle in fluid dynaics, Springer, Vienna, 006. Fundaentals and applications to agnetohydrodynaics, therodynaics, and astrophysics. [6] J. A. FONT, General relativistic hydrodynaics and agnetohydrodynaics: hyperbolic systes in relativistic astrophysics, in Hyperbolic probles: theory, nuerics, applications, Springer, Berlin, 008, pp [7] S. GALTIER, S. V. NAZARENKO, A. C. NEWELL, AND A. POUQUET, A weak turbulence theory for incopressible MHD, 000. [8] J.-F. GERBEAU, C. LE BRIS, AND T. LELIÈVRE, Matheatical ethods for the agnetohydrodynaics of liquid etals, Nuerical Matheatics and Scientific Coputation, Oxford University Press, Oxford, 006. [9] J. F. GIBSON, J. HALCROW, AND P. CVITANOVIĆ, Equilibriu and travelling-wave solutions of plane Couette flow, Journal of Fluid Mechanics, , pp [0] P. GOLDREICH AND S. SRIDHAR, Toward a theory of interstellar turbulence. : Strong alfvenic turbulence, Astrophysical Journal, , pp

23 [] M. GUNZBURGER AND A. LABOVSKY, High accuracy ethod for turbulent flow probles, M3AS: Matheatical Models and Methods in Applied Sciences, doi: 0./S [] J. G. HEYWOOD AND R. RANNACHER, Finite-eleent approxiation of the nonstationary Navier-Stokes proble. IV. Error analysis for second-order tie discretization, SIAM J. Nuer. Anal., 7 990, pp [3] HIDETOSHI AND HASHIZUME, Nuerical and experiental research to solve MHD proble in liquid blanket syste, Fusion Engineering and Design, 8 006, pp [] W. HILLEBRANDT AND F. KUPKA, eds., Interdisciplinary aspects of turbulence, vol. 756 of Lecture Notes in Physics, Springer- Verlag, Berlin, 009. [5] P. S. IROSHNIKOV, Turbulence of a conducting fluid in a strong agnetic field, Soviet Astrono. AJ, 7 96, pp [6] S. Y. KADIOGLU, R. KLEIN, AND M. L. MINION, A fourth-order auxiliary variable projection ethod for zero-mach nuber gas dynaics, J. Coput. Phys., 7 008, pp [7] R. KRAICHNAN, Inertial-range spectru of hydroagnetic turbulence, Physics of Fluids, 8 965, p [8] L. D. LANDAU AND E. M. LIFSHITZ, Electrodynaics of continuous edia, Course of Theoretical Physics, Vol. 8. Translated fro the Russian by J. B. Sykes and J. S. Bell, Pergaon Press, Oxford, 960. [9] W. LAYTON, H. TRAN, AND C. TRENCHEA, Stability of partitioned ethods for agnetohydrodynaics flows at sall agnetic Reynolds nuber, in Recent Advances in Scientific Coputing and Applications: Proceedings of the 8th International Conference on Scientific Coputing and Applications, E. M. Jichun Li and H. Yang, eds., vol. 586, AMS Conteporary Matheatics, 03. [30] W. LAYTON AND C. TRENCHEA, Stability of two IMEX ethods, CNLF and BDF-AB, for uncoupling systes of evolution equations, Applied Nuerical Matheatics, 6 0, pp. 0. [3] T. LIN, J. GILBERT, R. KOSSOWSKY, AND P. S. U. S. COLLEGE., Sea-Water Magnetohydrodynaic Propulsion for Next- Generation Undersea Vehicles, Defense Technical Inforation Center, 990. [3] E. MARSCH, Turbulence in the solar wind, in Reviews in Modern Astronoy, G. Klare, ed., Springer, Berlin, 990, p. 3. [33] M. L. MINION, Sei-iplicit spectral deferred correction ethods for ordinary differential equations, Coun. Math. Sci., 003, pp [3], Sei-iplicit projection ethods for incopressible flow based on spectral deferred corrections, Appl. Nuer. Math., 8 00, pp Workshop on Innovative Tie Integrators for PDEs. [35] B. PUNSLY, Black hole gravitohydroagnetics, vol. 355 of Astrophysics and Space Science Library, Springer-Verlag, Berlin, second ed., 008. [36] P. H. ROBERTS, An introduction to agnetohydrodynaics, Elsevier, 967. [37] J. V. SHEBALIN, W. H. MATTHAEUS, AND D. MONTGOMERY, Anisotropy in MHD turbulence due to a ean agnetic field, Journal of Plasa Physics, 9 983, pp [38] S. SMOLENTSEV, R. MOREAU, L. BÜHLER, AND C. MISTRANGELO, MHD therofluid issues of liquid-etal blankets: Phenoena and advances, Fusion Engineering and Design, 85 00, pp [39] R. TEMAM, Navier-Stokes Equations. Theory and Nuerical Analysis, North-Holland, Asterda, 979. [0], Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Matheatics, Society for Industrial and Applied Matheatics, Philadelphia, 995. [] C. TRENCHEA, Unconditional stability of a partitioned IMEX ethod for agnetohydrodynaic flows, tech. rep., University of Pittsburgh, 0. [] M. K. VERMA, Statistical theory of agnetohydrodynaic turbulence: Recent results, Physics Reports, 0 00, p. 9. [3] I. VESELOVSKY, Turbulence and waves in the solar wind foration region and the heliosphere, Astrophys. Space Sci., 77 00, pp. 9. 3