Athena中算法与GLM九个方程

Athena中算法与中算法与GLM九个方程九个方程刘静静刘静静4 4月月7 7号号主要内容主要内容n n阶段任务n n背景(CESE+HLL)n nAthena中的不同算法n nGLM-MHD九个方程n n参考文献阶段任务阶段任务对程序对程序CESECESE（in the near in the near sun)andsun)and HLL(offHLL(off-sun)-sun)的一些修改，从以下三个方的一些修改，从以下三个方面：面：n n在在off-sunoff-sun部分使用参考部分使用参考Athena codeAthena code中的不同算法试验中的不同算法试验n n两部分使用不同的参数无量纲化，在两部分使用不同的参数无量纲化，在off-sunoff-sun部分使用部分使用6Rs6Rs处的太阳分参处的太阳分参数无量纲化数无量纲化n n考虑将考虑将MHDMHD方程组改成方程组改成GLMGLM九个方程，模拟太阳风九个方程，模拟太阳风背景背景(CESE+HLL）n n从太阳到地球的计算区域，分成了近太阳和远太阳两部分区域从太阳到地球的计算区域，分成了近太阳和远太阳两部分区域n n在近太阳区域，使用阴阳网格和在近太阳区域，使用阴阳网格和CESECESE算法，在远太阳区域使用自适应算法，在远太阳区域使用自适应（AMRAMR）网格和）网格和HLLHLL算法算法n n阴阳网格可以避免奇点和在极区网格集中问题，更好地在太阳表面达到高精度阴阳网格可以避免奇点和在极区网格集中问题，更好地在太阳表面达到高精度的解；的解；AMRAMR可以自动的捕捉等离子体流的特征（可以自动的捕捉等离子体流的特征（far-field)far-field)，例如日球层电流片，例如日球层电流片和激波，并且可以节省计算资源和激波，并且可以节省计算资源n n控制方程：控制方程：其中 ,磁场分成背景磁场和扰动磁场AthenaAthenaAthena：n nThe equations:ideal MHDThe equations:ideal MHDn nThe numerical algorithms in The numerical algorithms in AthnaAthna are based on directionally are based on directionally unsplit,higherunsplit,higher order order Godunov methodsGodunov methodsn nDiscretizationDiscretization 1)mass,momentum,energy:finite volume 1)mass,momentum,energy:finite volume 2)magnetic 2)magnetic field:basedfield:based on area rather than volumes averages on area rather than volumes averagesVolume-averagedtime-and area-averaged fluxeswithAthenawitharea-averagedelectromotive force averaged along the appropriate line elementn advantages:1)ideal for use AMR2)Superior for shock capturing and evolving the contact and rotational discontinutiesAthnaThe chart for the steps in the 2D algorithm in AthenaAthenan nThe algorithm for computing MHD interface The algorithm for computing MHD interface states:piecewisestates:piecewise contant(firstcontant(first-order)-order)reconstruction,piecewisereconstruction,piecewise linear(second-order)resconstruction,piecewiselinear(second-order)resconstruction,piecewise parabolic(thirdparabolic(third-order)reconstructionorder)reconstructionn nThe algorithm for computing The algorithm for computing fluxes:HLLfluxes:HLL solvers,Roessolvers,Roes method methodRemarksRemarks：1)the reconstruction used in Athena require characteristic variables and a 1)the reconstruction used in Athena require characteristic variables and a characte-risticcharacte-ristic evolution of the evolution of the linearizedlinearized systermsysterm 2)The Godunov methods do not require expensive solvers based on complex characteristic 2)The Godunov methods do not require expensive solvers based on complex characteristic decompositionsdecompositionsData reconstructionn nPiecewise constant reconstruction:assume the primitive variables are piecewise Piecewise constant reconstruction:assume the primitive variables are piecewise constant within each cellconstant within each celln nPiecewise linear reconstruction:assume the primitive variables vary linearly within Piecewise linear reconstruction:assume the primitive variables vary linearly within each celleach cellwhere is a limited slope for the cell,two types of limiter as follow:n WENO reconstruction:can achieve higher than second orderBasic ideal:several cells can formulate a module (r denotes the number of cells formulated the module,k denotes the total number of modules,different modules have different interpolation polynomialsData reconstructionn nthe total polynomial the total polynomial R(xR(x)of reconstruction is a convex combination of the)of reconstruction is a convex combination of the above polynomials above polynomials Pj(xPj(x),),where is weight cofficient,The WENO reconstruction can have(2k-1)order,and is non-oscillatory,but the computation is complexGodunov Fluxesn nFirst proposed by First proposed by GodunovGodunov S.K.in 1959 S.K.in 1959n nThe basic idea:at ,in each cell the primitive variables are constant.At the The basic idea:at ,in each cell the primitive variables are constant.At the interface interface bewteenbewteen the neighbor cells ,there is a initial the neighbor cells ,there is a initial discontinuityn nGodunovGodunov methods do require expensive solvers based on complex characteristic methods do require expensive solvers based on complex characteristic decompositions and capture high quality shock decompositions and capture high quality shock n nHLL-family solvers:HLL-family solvers:then formulate a local Riemann problem bewteen the neighbour cells.Roes methodn nAn useful linearization for the MHD equationsAn useful linearization for the MHD equationsn nInclude all the characteristics of the Include all the characteristics of the systerm,andsysterm,and less diffusive and more accurate less diffusive and more accurate for intermediate wavesfor intermediate wavesn nJacobianJacobian is evaluated using an average is evaluated using an average state(Roestate(Roe average)average)where is the enthalpywhere is the enthalpy the Roe fluxes are simply:the Roe fluxes are simply:Disadvantage:may return negative densities or pressuresDisadvantage:may return negative densities or pressuresHLLnassuming an average intermediate state between the fastest and slowest wavesnintermediate statenthe HLL fluxes are the minimum signal speed and the maximum signal speedHLLRemarks:Remarks:n nmust be estimated appropriatelymust be estimated appropriatelyDavis Davis EinfeldtEinfeldt et al et aln The solver is fast and do not need the characteristic decompositionntoo diffusive and cannot resolve isolated contact discontinuities very wellHLLEn nUsing a Using a singalsingal constant intermediate state computed from a conservative average constant intermediate state computed from a conservative averagen nDo not require a characteristic decomposition of MHD equationsDo not require a characteristic decomposition of MHD equationsn nThe HLLE flux at the interface:The HLLE flux at the interface:whereare the fluxes evaluated using the left andright states of the conserved variables,andIf both (or ),the HLLE flux will be n the HLLE can guarantee the pressure and density is positive,but in the multiple dimensions,it does not necessarily guarantee.Whats more,the HLLE neglects the Alfven,slow magnetosonic,and contact waves.HLLCnthe intermediate states in the Riemann fan are separated into two intermediate states by a contact discontinuity can resolve isolated contact discontinuities exactlybe evaluated from HLLaveragen the numerical flux of HLLCHLLCn nPositively conservativePositively conservativen nHLLC can dramatically improve the results of the HLL solver,and has much less HLLC can dramatically improve the results of the HLL solver,and has much less computational time than the HLLEM computational time than the HLLEM a:the soud speed HLLDn nFive-wave Riemann solver for Five-wave Riemann solver for MHD,HLL-Discontinuities solverMHD,HLL-Discontinuities solvern nComposed of four intermediate Composed of four intermediate statesstates indicate the speeds of the fast magnetosonic waves,Alfvn waves,and entropy wave HLLDn nThe numerical flux vector of the HLLD Riemann solver for MHD equationsThe numerical flux vector of the HLLD Riemann solver for MHD equationsn n Positively conservative Positively conservative:the fast magnetosonic speed两部分无量纲化两部分无量纲化n n在在off-sunoff-sun部分使用部分使用6Rs6Rs处太阳风参数做无量纲化处太阳风参数做无量纲化CartesianCartesian部部分，利用初始时的流场和磁场参数分，利用初始时的流场和磁场参数ParkerParker解在解在6Rs6Rs处的值处的值 Call parkerCall parker（r6rs,ur,gamma0,T0)r6rs,ur,gamma0,T0)n n进一步改进：在给定的网格上，指定半径处的太阳风参数，进一步改进：在给定的网格上，指定半径处的太阳风参数，分两部分无量纲化分两部分无量纲化 subroutine subroutine nearpoint(rnearpoint(r)距离最小的点subroutine nearpoint(r)给定的半径利用Parker解求出所得到点的太阳风参数，进行无量纲化GLM-MHDn nGLM(generalized Lagrange multiplier)n nThe form of equationsn nSolver for GLM-MHDGLMn nCoupling the divergence constraint by introducing a generalized Lagrange Coupling the divergence constraint by introducing a generalized Lagrange multipliermultipliern nthe divergence errors are transported to the domain boundaries with the the divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same timemaximal admissible speed and are damped at the same timen nMagnetic induction equations are replaced:Magnetic induction equations are replaced:Different choices for the linear operator DElliptic correction:Elliptic correction:Parabolic correction:Parabolic correction:Hyperbolic correction:Hyperbolic correction:Combination of parabolic and hyperbolic Combination of parabolic and hyperbolic ansatzansatz:方程组形式方程组形式n nMHD方程组变为方程形式方程形式n nThe MHD equations can beThe MHD equations can be symmetrizedsymmetrized by adding some by adding some hyperbollichyperbollic terms on the terms on the right-hand sideright-hand siden nThe changed The changed eqationseqations:Remarks:1)call the equations the extended GLM(EGLM)formulation of MHD equations 2)significantly depends on the grid size and the scheme used,is a function of 方程的特征值方程的特征值n nThe The eigenvalueseigenvalues of the GLM-MHD of the GLM-MHD coinsidecoinside with the ordinary MHD waves plus with the ordinary MHD waves plus two additional modes ,for a total of nine characteristic wavestwo additional modes ,for a total of nine characteristic wavesn nFor one dimensional,x directionFor one dimensional,x directionremarks：1)show that the system is hyperbolic 2)only the waves traveling with speeds can carry a change inor The solver of GLM-MHDn nSolver for the GLM-MHD without additional sourceSolver for the GLM-MHD without additional sourcen nTreat the linear system given by the B and from the other ordinary 7-Treat the linear system given by the B and from the other ordinary 7-wave MHD equations in an operator-split fashionwave MHD equations in an operator-split fashionwhere S and A are the advection and source step operators separately1)Advection step:based on the corner transport upwind(CTU)method,second order accurate discretizationWhere F,G,H are the numerical fluxes computed by solving a Riemann problem between suitable time-centered left and right statesR(,)denotes the flux obtained by means of a Riemann solver,are computed via a Taylor expansion in the direction normal to a given interfaceThe solver for GLM-MHD2)Source step:solver the initial value problem without the term2)Source step:solver the initial value problem without the term can be integrated exactly for a time incrementRemarks:n numerical experiments indicate that the divergence errors are mininized when the lies in the range 0,1n is an unphysical variable,the initial condition given by the output of the most recent stepnBoundary condition for :assume that the behavior of and at the boundary is identical,use a homogeneous Dirichlet condition,nonreflecting boundary condition参考文献参考文献n nXueshangXueshang Feng,ShaohuaFeng,Shaohua Zhang,ChangqingZhang,Changqing Xiang,LipingXiang,Liping Yang,ChaoweiYang,Chaowei Jiang,”AJiang,”A Hybrid Solar Wind Model of CESE+HLL Method with Yin-Yang Overset Grid and Hybrid Solar Wind Model of CESE+HLL Method with Yin-Yang Overset Grid and AMR Grid”AMR Grid”n nTakahiro Takahiro Miyoshi,NaokiMiyoshi,Naoki Terada,”TheTerada,”The HLLD Approximate Riemann Solver for HLLD Approximate Riemann Solver for MagnetosphericMagnetospheric Simulation”Simulation”n nTakahiro Takahiro Miyoshi,KanyaMiyoshi,Kanya Kusano,”AKusano,”A multi-state HLL approximate solver for ideal multi-state HLL approximate solver for ideal magnetohydrodynamicsmagnetohydrodynamics”n nA.Mignone,G.Bodo,”PLUTO:AA.Mignone,G.Bodo,”PLUTO:A NUMERICAL CODE FOR COMPUTATIONAL NUMERICAL CODE FOR COMPUTATIONAL ASTROPHYSICS”ASTROPHYSICS”n nShengtaiShengtai Li,”AnLi,”An HLLC Riemann solver for magneto-hydrodynamics”HLLC Riemann solver for magneto-hydrodynamics”n nJames James M.Stone,ThomasM.Stone,Thomas A.Gardiner,”ATHENAA.Gardiner,”ATHENA：A NEW CODE FOR A NEW CODE FOR ASTROPHYSICAL MHD”ASTROPHYSICAL MHD”n nA.Dedner,F.Kemm,”HyperbolicA.Dedner,F.Kemm,”Hyperbolic Divergence Cleaning for the MHD Equations”Divergence Cleaning for the MHD Equations”n nAndrea Andrea Mignone,PetrosMignone,Petros Tzeferacos,GianluigiTzeferacos,Gianluigi Bodo,”HighBodo,”High-order conservative finite-order conservative finite difference GLM-MHD schemes for cell-centered MHDdifference GLM-MHD schemes for cell-centered MHD参考文献参考文献n nAndrea Andrea Mignone,PetrosMignone,Petros Tzeferacos,”ATzeferacos,”A Second-Order Second-Order UnsplitUnsplit GodunovGodunov Scheme for Scheme for Cell-Cell-CenteresCenteres MHD:CTU-GLM scheme”MHD:CTU-GLM scheme”n nShengtaiShengtai Li,HuiLi,Hui Li,”ALi,”A Modern Code for Solving Magneto-hydrodynamics or Hydro-Modern Code for Solving Magneto-hydrodynamics or Hydro-dynamics Equationsdynamics Equationsn nDinshawDinshaw S.Balsara,”MultidimensionalS.Balsara,”Multidimensional HLLE Riemann HLLE Riemann Solver;ApplicationSolver;Application to Euler and to Euler and MagnetohydrodynamicMagnetohydrodynamic Flows”Flows”