Athena中算法与GLM九个方程PPT课件

Athena中算法与中算法与GLM九个方程九个方程刘静静4月7号主要内容主要内容n阶段任务n背景(CESE+HLL)nAthena中的不同算法nGLM-MHD九个方程n参考文献阶段任务阶段任务对程序CESE（in the near sun)and HLL(off-sun)的一些修改，从以下三个方面：n在off-sun部分使用参考Athena code中的不同算法试验n两部分使用不同的参数无量纲化，在off-sun部分使用6Rs处的太阳分参数无量纲化n考虑将MHD方程组改成GLM九个方程，模拟太阳风背景背景(CESE+HLL）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n在近太阳区域，使用阴阳网格和CESE算法，在远太阳区域使用自适应（AMR）网格和HLL算法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的解；AMR可以自动的捕捉等离子体流的特征（far-field)，例如日球层电流片和激波，并且可以节省计算资源n控制方程：BBuBBUuBBBBurgujEBBBBBuuBBBBBurgBjBBBBBBBBBuuuu01 ttetttQtpeep)()()(0110110121001001110121212210其中 ,磁场分成背景磁场和扰动磁场00BjBuE,10BBBAthenaAthena：nThe equations:ideal MHDnThe numerical algorithms in Athna are based on directionally unsplit,higher order Godunov methodsnDiscretization 1)mass,momentum,energy:finite volume 2)magnetic field:based on area rather than volumes averagesdxdydttzyxHyxtHdxdzdttzyxGzxtGdydzdttzyxFzytFnnjjiinnkkiinnkkjjttyyxxknkjittzzxxjnkjittzzyyinkji 121212121121212121121212121,1,1,1212121,2121,21,2121,21Volume-averageddxdydztzyxUzyxUkkjjiizzyyxxnnkji212121212121,1,time-and area-averaged fluxes2121,2121,21,21,21,21,21,2121,21,1,nkjinkjinkjinkjinkjinkjinkjinkjiHHztGGytFFxtUUwithAthena2121,21,2121,21,21,21,21,21,21,21,1,21,nkjiynkjiynkjiznkjiznkjixnkjixztytBBdydztzyxBzyBnizzyyxnkjixkkjj,121,21,21212121withdxdttzyxtxkjttxxxnkjixnnii,121212121,21,12121 area-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 AthenaAthenanThe algorithm for computing MHD interface states:piecewise contant(first-order)reconstruction,piecewise linear(second-order)resconstruction,piecewise parabolic(third-order)reconstructionnThe algorithm for computing fluxes:HLL solvers,Roes methodRemarks：1)the reconstruction used in Athena require characteristic variables and a characte-ristic evolution of the linearized systerm 2)The Godunov methods do not require expensive solvers based on complex characteristic decompositionsData reconstructionnPiecewise constant reconstruction:assume the primitive variables are piecewise constant within each cellnPiecewise linear reconstruction:assume the primitive variables vary linearly within each celliiRiiLuuuu21121,iiiiuxxxuxuwhereiu is a limited slope for the cell,two types of limiter as follow:21211111,modminiiiiiiiuuuuuuu11212111212iiiiiiiiiiiuuuuuuuuuuun 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 polynomialsrkS xPkData reconstructionnthe total polynomial R(x)of reconstruction is a convex combination of the above polynomials Pj(x),xPwxRkjjj10where is weight cofficient,jw1010kjjjww,The WENO reconstruction can have(2k-1)order,and is non-oscillatory,but the computation is complexGodunov FluxesnFirst proposed by Godunov S.K.in 1959nThe basic idea:at ,in each cell the primitive variables are constant.At the interface bewteen the neighbor cells ,there is a initial discontinuitynGodunov methods do require expensive solvers based on complex characteristic decompositions and capture high quality shock nHLL-family solvers:nt11iiiixxandxx,iniiniixxuxxuxu,2121then formulate a local Riemann problem bewteen the neighbour cells.0 0 0 11NRkkkLififif,fffFRoes methodnAn useful linearization for the MHD equationsnInclude all the characteristics of the systerm,and less diffusive and more accurate for intermediate wavesnJacobian is evaluated using an average state(Roe average)where is the enthalpy the Roe fluxes are simply:Disadvantage:may return negative densities or pressures RLRRLLRLRRLLRLRRLLRLBBBHHHvvvPEHkkRLkkRLRoeRUULffF21HLLnassuming an average intermediate state between the fastest and slowest wavesnintermediate statenthe HLL fluxesRLSS,are the minimum signal speed and the maximum signal speedLRLRLLRRSSFFUSUSU0S if FS0S if F 0S RRRLLifFFLHLLLRLRLRRLLRSSUUSSFSFSFHLLRemarks:nmust be estimated appropriatelyDavis Einfeldt et alRmLmLRlLlRUUSUUS,min,maxRmRoemLRoelLlRUUSUUS,min,maxn The solver is fast and do not need the characteristic decompositionntoo diffusive and cannot resolve isolated contact discontinuities very wellHLLEnUsing a singal constant intermediate state computed from a conservative averagenDo not require a characteristic decomposition of MHD equationsnThe HLLE flux at the interface:1212121iiiRiLHLLEibbbbbbbbuuffF,where21212121iRiRiLiL,uffuffare the fluxes evaluated using the left andright states of the conserved variables,and00,minmin,maxmaxLRfLlfRMcvbcvbIf both (or ),the HLLE flux will be 0 and 0RfRMcv0 and 0LfLLcv2121 iRiLor,ffn 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 exactlyLnLLRnRRLRLnLLRnRRnLLnLLnRRnRRnMvSvSpTpTvSvSvvSvvSvS11be evaluated from HLLaverageMSn the numerical flux of HLLC0S if 0S if 0S if 0S if RMLLRRRMLLHLLCFSFSFFFRRLRRLLLLLUUSFFUUSFFHLLCnPositively conservativenHLLC can dramatically improve the results of the HLL solver,and has much less computational time than the HLLEM RRRLLLauSauS21,21a:the soud speed pHLLDnFive-wave Riemann solver for MHD,HLL-Discontinuities solvernComposed of four intermediate statesMSSS,indicate the speeds of the fast magnetosonic waves,Alfvn waves,and entropy wave nLnRnLnRfnLfnRLfnLfnRRcvcvScvcvS,min,maxLnMLRnMRBSSBSS,LnLLRnRRLRLnLLRnRRnLLnLLnRRnRRnMvSvSpTpTvSvSvvSvvSvS11HLLDnThe numerical flux vector of the HLLD Riemann solver for MHD equationsRLfRRfLLcuScuS21,21:the fast magnetosonic speedRLffcc,0S if 0S if 0S if 0S if 0S if 0S if RRMLLLRRRLLLnRnRnMnLnnHLLCFSFSFSFSFFF两部分无量纲化两部分无量纲化n在off-sun部分使用6Rs处太阳风参数做无量纲化Cartesian部分，利用初始时的流场和磁场参数Parker解在6Rs处的值 Call parker（r6rs,ur,gamma0,T0)n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 subroutine nearpoint(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-MHDnGLM(generalized Lagrange multiplier)nThe form of equationsnSolver for GLM-MHDGLMnCoupling the divergence constraint by introducing a generalized Lagrange multipliernthe divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same timenMagnetic induction equations are replaced:00BDuBBuBTTtDifferent choices for the linear operator D 0D,0,12ppccD,0,12hthccD 2211pthccD方程组形式方程组形式nMHD方程组变为2222202100210)(phhttTTtTTtxxtccBcBuBuBpeeuBBuBBBBpuuuu 方程形式方程形式nThe MHD equations can be symmetrized by adding some hyperbollic terms on the right-hand sidenThe changed eqations:2222221210)(phhttTTtTTtxxtccuBcBuBBuBuBpeeuBuBBuBBBBBBpuuuu 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 pcpchc方程的特征值方程的特征值nThe eigenvalues of the GLM-MHD coinside with the ordinary MHD waves plus two additional modes ,for a total of nine characteristic wavesnFor one dimensional,x directionhcxsxaxfxhvcvcvcvc56,47,38,29,1,zfzyfyxfxhcvcvcvc,maxremarks：1)show that the system is hyperbolic 2)only the waves traveling with speeds can carry a change inor 91，xBThe solver of GLM-MHDnSolver for the GLM-MHD without additional sourcenTreat the linear system given by the B and from the other ordinary 7-wave MHD equations in an operator-split fashion22ntttnSASUU221where S and A are the advection and source step operators separately1)Advection step:based on the corner transport upwind(CTU)method,second order accurate discretizationzyxtnknknjnjnininnn2121212121212121212121211HHGGFFUUWhere F,G,H are the numerical fluxes computed by solving a Riemann problem between suitable time-centered left and right states211212121211212121211212121nknkninjnjnininini,R,R,RVVHVVGVVFR(,)denotes the flux obtained by means of a Riemann solver,are computed via a Taylor expansion in the direction normal to a given interface21121nknk,VVThe solver for GLM-MHD2)Source step:solver the initial value problem without the term B22phcctcan be integrated exactly for a time increment zyxhcchwiththcphhtn,min,exp 0Remarks: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 0参考文献参考文献nXueshang Feng,Shaohua Zhang,Changqing Xiang,Liping Yang,Chaowei Jiang,”A Hybrid Solar Wind Model of CESE+HLL Method with Yin-Yang Overset Grid and AMR Grid”nTakahiro Miyoshi,Naoki Terada,”The HLLD Approximate Riemann Solver for Magnetospheric Simulation”nTakahiro Miyoshi,Kanya Kusano,”A multi-state HLL approximate solver for ideal magnetohydrodynamics”nA.Mignone,G.Bodo,”PLUTO:A NUMERICAL CODE FOR COMPUTATIONAL ASTROPHYSICS”nShengtai Li,”An HLLC Riemann solver for magneto-hydrodynamics”nJames M.Stone,Thomas A.Gardiner,”ATHENA：A NEW CODE FOR ASTROPHYSICAL MHD”nA.Dedner,F.Kemm,”Hyperbolic Divergence Cleaning for the MHD Equations”nAndrea Mignone,Petros Tzeferacos,Gianluigi Bodo,”High-order conservative finite difference GLM-MHD schemes for cell-centered MHD参考文献参考文献nAndrea Mignone,Petros Tzeferacos,”A Second-Order Unsplit Godunov Scheme for Cell-Centeres MHD:CTU-GLM scheme”nShengtai Li,Hui Li,”A Modern Code for Solving Magneto-hydrodynamics or Hydro-dynamics EquationsnDinshaw S.Balsara,”Multidimensional HLLE Riemann Solver;Application to Euler and Magnetohydrodynamic Flows”