Uniformly robust preconditioners for incompressible MHD system

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Abstract

In this paper, we study efficient schemes and design robust preconditioners for solving incompressible MHD (magnetohydrodynamics) system. We propose the first order and second order symmetric schemes with augmented terms in magnetic equations that are introduced in consideration of designing uniformly robust preconditioners. We present optimal error estimates of the proposed time-matching scheme. Further, we design block diagonal preconditioners for the schemes, and rigorously prove that the condition number of preconditioned system is uniformly bounded by a constant which only depends on the domain Ω. Finally, some numerical experiments, including accuracy tests and physical benchmark problems, are presented to verify the accuracy of the schemes and the robustness of the preconditioner.

Introduction

We consider the following incompressible d(d=2,3) dimensional MHD equations under the influence of the body force and applied currents: utνΔu+(u)u+p+sB×curlB=f,divu=0,Bt+ηcurlcurlBcurl(u×B)=g,divB=0,for (x,t)Ω×(0,T) with ΩRd and a fixed T(0,). Here, Ω is a connected, bounded domain which is either convex or has a C1,1 boundary Ω, u denotes the velocity field, p is the pressure, B is the magnetic field, f is a known body force, and g is a known applied current with divg=0 and gn|Ω=0, where n denotes the outer unit normal of Ω. The physical parameters ν1=Re (fluid Reynolds number, we assume Re1), η1=Rm (magnetic Reynolds number) and the coupling coefficient s are given by Re=ULμf,Rm=μmσUL,s=B2ρμmU2,where U is the characteristic velocity, L the characteristic length, μf the kinematic viscosity, μm the magnetic permeability, σ the electric conductivity, B the characteristic magnetic field, ρ the fluid density. The system is considered in conjunction with the following boundary and initial conditions u|Ω=0,Bn|Ω=0,n×curlB|Ω=0,u(x,0)=u0(x),B(x,0)=B0(x) with divu0(x)=0, divB0(x)=0.

The MHD equations model the dynamic behaviors of an electrically conducting fluid under the effect of an imposed magnetic field. Its applications involve in geophysics, astrophysics, fusion reactor blankets and confinement for controlled thermonuclear fusion, see [1], [2], [3]. The governing equations behind MHD couple the Navier–Stokes equations for hydrodynamics and Maxwell’s equations for electromagnetism. The two types equations are coupled by the Lorentz force, which governs the effect of a magnetic field on fluid flow, and the appearance of the fluid velocity in Ohm’s law, which accounts for the influence of hydrodynamics on the electric current. Concerning the corresponding extensive theoretical modeling/numerical analysis of the MHD system, we refer to [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and the references therein.

In recent years, there have been many numerical investigations for the MHD system and other corresponding works such as [22], [23], [24], [25], [26], [27]. For example, in [8], [28], optimal error estimates for semi-implicit scheme of the MHD system were proved, and numerical performances were also presented. In [29], [30], convergence analyses of Crank–Nicolson extrapolation scheme were performed. In [31], some coupled and decoupled schemes for approximating weak solution and regular strong solution were proposed and analyzed. In [32], stability and convergence of a BFF2 linear scheme were investigated. In [9], two partitioned schemes for a reduced MHD system at small magnetic Reynolds number were proposed and analyzed. In [33], under low regular assumption of the solutions and domain geometries, an optimal error estimate for the semi-implicit scheme was established. In [34], [35], [36], [37], [38], stability and convergence of decoupled schemes for the MHD problems were discussed. In [39], convergence analysis was carried out for the MHD model in a nonconvex, nonsmooth and multiconnected domain. In [40], the author developed a parallel, scalable multigrid-based and physics-based preconditioning strategy for the compressible MHD equations. Numerical tests demonstrate the excellent scalability properties of the solver in terms of grid refinement. In [41], the authors employed Newton–Krylov methods that are preconditioned by fully-coupled algebraic multilevel preconditioners for the MHD problem. They verified the preconditioner is robust and scalable. In [5], [42], the authors studied the block preconditioners by approximate block factorization approach for the MHD system, and demonstrated the robustness of these preconditioners with respect to mesh refinement. In [43], the authors designed block LU preconditioners for the thermally coupled MHD equations with good properties with respect to the mesh size and the Hartmann number. In [16], the authors developed block preconditioners for a new MHD model that preserves divergence free of magnetic field. They proved the preconditioners are robust with respect to most physical and discretization parameters. In [44], the authors proposed a Newton–Krylov solver for stationary MHD equations. They demonstrated the robustness of the proposed solver with respect to mesh refinement.

Seen from the above literatures, in the numerical studies for the MHD problems, there mainly includes two respects: (i) proposing efficient schemes and carrying out numerical analysis and (ii) designing robust solvers for solving the linear systems. In the most existing literatures, people often focus on one respect. In this paper, we attempt to connect the two respects. We propose the first order and second order time-marching schemes for the MHD equations and provide their optimal error estimates. Then, we design block diagonal preconditioners for the schemes using the operator preconditioning method [45], [46] and prove their robustness.

In most of MHD solvers literatures, such as [5], [40], [41], [42], [43], [44], people often focus on the robustness of the preconditioners with respect to mesh refinement, and pay a little attention to the robustness on the physical parameters. In [16], the authors designed preconditioners that are robust with respect to most physical parameters. They demonstrated the condition number of the preconditioned system depends on the coupling coefficient s and the numerical solution of magnetic field Bhn. In this paper, we aim to study uniformly robust preconditioners for the MHD problem. In order to avoid such a condition like [16], we subtly introduce a symmetric term in magnetic equations, and define an appropriate block diagonal preconditioner for the augmented system. We rigorously prove the condition number of the preconditioned operator is uniformly bounded by a constant which only depends on the domain Ω. Thus, the proposed preconditioner is robust with respect to mesh size h, time step size δt, and physical parameters Re,Rm,s. We also present the L2 and H1 optimal error estimates of the velocity field u and magnetic field B, and the L2 optimal error estimates of the pressure p for the time-marching scheme. Finally, numerical experiments, including convergence tests and physical benchmark problems, are implemented to test the accuracy of our schemes and validate uniform robustness of the proposed preconditioner.

The rest of paper is organized as follows. In Section 2, we propose the first order time-marching scheme and provide its optimal error estimates. In Section 3, we design a block diagonal preconditioner for the scheme and prove its uniform robustness. In Section 4, we extend the discussions to the second order schemes. In Section 5, ample numerical experiments are presented to test our theoretical results. Finally, some concluding remarks are given in Section 6.

Section snippets

First order scheme and error estimates

In this section, we propose the first order time-marching scheme for problem (1.1) and demonstrate error estimates. We first define some notations of function spaces and norms. For two vector functions x,y, we denote the L2 inner product as (x,y)=Ωxydx and L2 norm x2=(x,x). For function setting of this MHD model, we also define several standard Sobolev spaces: H01(Ω)=ϕH1(Ω)d:ϕ|Ω=0,L02(Ω)=ϕL2(Ω):Ωϕdx=0,Hn1(Ω)={ϕH1(Ω)d:nϕ|Ω=0}, and use 1 for the norm in H1(Ω)d. In H01(Ω) and Hn1(Ω)

Robust preconditioner

In this section, we study the robust preconditioner for the scheme (2.3). We first present spatial discretization for (2.3) based on finite element method. For l1, r1, we define the conforming finite element spaces VhH01(Ω),QhL02(Ω),ChHn1(Ω) which consist of continuous piecewise polynomials of degree l,l1 and r, respectively. Moreover, we assume Vh and Qh satisfy the inf–sup condition: βqhsupvhVh(divvh,qh)vh,qhQh,where the constant β only depends on the domain Ω. A well known

Second order schemes and robust preconditioners

In this section, we generalize our discussions to second order time-marching schemes. With the notations Dtwn=3wn4wn1+wn22δt,ŵn1=2wn1wn2,w̄n=wn+wn12,w˜n1=32wn112wn2,the two second order schemes based on BDF2 and Crank–Nicolson methods are presented as follows.

Second order scheme I(BDF2): Find unH01(Ω),pnL02(Ω),BnHn1(Ω) satisfying (Dtun,v)+ν(un,v)pn,divv(divun,q)+s(B̂n1×curlBn,v)=(fn,v)(ûn1ûn1,v),(DtBn,C)η(curlBn,curlC)η(divBn,divC)+(un×B̂n1,curlC)δts(B̂n1×curlBn,

Numerical examples

In this section, we present accuracy tests and benchmark problems to test the accuracy of the schemes and the robustness of the proposed preconditioner (3.13). Here, we verify the robustness of the preconditioner (3.13) for the first order scheme (3.5). We expect the preconditioner (4.3) for the second order schemes has the similar performance. For spatial discretization, we use conforming finite elements (P2,P1,P2) to discretize velocity, pressure and magnetic field for island coalescence

Conclusion remarks

In this paper, we propose the first order and second order time-marching schemes and devise robust preconditioners for solving the MHD problem. We provide the optimal error estimates for the proposed time-marching scheme. Furthermore, we design a block diagonal preconditioner for the proposed scheme using the operator preconditioning framework, and rigorously prove that the condition number of preconditioned system is bounded by a constant that only depends on domain Ω. Numerical experiments

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    This work is partially supported by National Natural Science Foundation of China under Grant Numbers 11601468 and 11771375, and Shandong Province Natural Science Foundation, China (ZR2018MA008).

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