Changes between Version 15 and Version 16 of AstroBearProjects/resistiveMHD
- Timestamp:
- 08/24/11 22:49:25 (13 years ago)
Legend:
- Unmodified
- Added
- Removed
- Modified
-
AstroBearProjects/resistiveMHD
v15 v16 44 44 = ====================== = 45 45 46 Here I overview fourapproaches to solve the resistive MHD equations. [[BR]][[BR]]46 Here I overview two approaches to solve the resistive MHD equations. [[BR]][[BR]] 47 47 (1)Solving the decomposed format with divergence cleaning (Implicit) [[BR]] 48 48 This is the simplest method in terms of solving scheme. This method is based on the simplification that in some cases eta is a constant in the region of interest. This will simplify the above equation into the following form:[[BR]] … … 52 52 This equation holds true for each separate field component, and the form resembles the linear isotropic thermal conduction equation. We can solve the elliptic equation for three times or we can write a expanded linear system so that the solver is only called once: [[BR]] 53 53 54 [[ matrix1]][[BR]]54 [[Image(http://www.pas.rochester.edu/~shuleli/rmhd_matrix1.png, 50%]][[BR]] 55 55 56 56 In the constant diffusivity case, the matrix M are identical for the Bx, By and Bz components. Define C=[-6k,k,k,k,k,k,k], where [[latex($k = \eta dt/dx^2$)]]. One can immediately recognize that k is the diffusion CFL number. Then the expanded matrix is still "tridiagonal" in a 3-D sense and can be blocked into three identical parts: [[latex($M_i=C-[1,0,0,0,0,0,0]$)]] on each row. The RHS vector are source terms that can be easily obtained by the following the CR scheme:[[BR]] … … 62 62 The field obtained above resides at cell centers. We can do interpolations to retrieve the new magnetic field at each cell faces. But since we are treating the x, y and z fields independently and applying sources that is not necessarily divergence free to them, we are running the risk of introducing divergence to the face centered magnetic field components which should be divergence free. Experiments show that this decomposing method can introduce large divergence to the face centered field. Thus a divergence cleaning scheme is required once we finish the diffusion update. Here we introduce the divergence cleaning scheme implemented in AstroBEAR. Considering a cell with face centered field components:[[BR]] 63 63 64 [[ dia1]][[BR]]64 [[[[Image(http://www.pas.rochester.edu/~shuleli/rmhd_dia1.png, 50%]][[BR]]]][[BR]] 65 65 66 66 We can find the divergence source term: [[latex($d=\nabla \cdot B$)]]. It is obvious that if B is face centered, then its divergence should be cell centered.[[BR]] … … 95 95 Here we treat the face centered field components directly by looking at the following diagram: [[BR]] 96 96 97 [[ dia2]]97 [[[[Image(http://www.pas.rochester.edu/~shuleli/rmhd_dia2.png, 50%]][[BR]]]] 98 98 99 99 We use Bz at the top face of the cell as an example for demonstration. Here, the changing of Bz does not only depend on the Bz alone, but also the field components of Bx and By. The second order accurate scheme requires 12 closest points surrounding the face center. In the diagram, F denotes face centers, X denotes edge centers. The face centers F1 to F8 have distance of [[latex($sqrt{1/2}$)]], and the face centers F9 to F12 have distance of 1. A larger lattice cell can be considered using further face centers when a higher accuracy is required. The correlation vector C in this case has 13 elements which relates the face center we are considering F0 to the 12 adjacent face centers F1 to F12. Its elements can be written as: [[BR]] … … 117 117 The matrix equation has the form of: [[BR]] 118 118 119 [[ matrix2]]119 [[[[Image(http://www.pas.rochester.edu/~shuleli/rmhd_matrix2.png, 50%]][[BR]]]] 120 120 121 121 where M_x, M_y and M_z are the diagonal contributions from the face centers F9 to F12: M = [C_0, C_9, C_10, C_11, C_12, ....]-[1,0,0,0...]; and the blocked M is different for x, y and z components as stated before. There are also nondiagonal blocks which connect Bx block to By and etc. These are marked by D, which comes from the C_1 to C_8 contributions. The source vector is obtained by construct the matrix with -C_0-1; C1; C2; ... C12 and dot it with the field vector: [Bx1 ... Bxn; By1 ... Byn; Bz1 ... Bzn]. The scheme involves looking at each face centers and find out it C vector. Once the C vector is obtained, it can be used to construct the matrix and the source vector. [[BR]] 122 122 This matrix equation is still solvable using linear solver package in parallel, but the construction of the linear system is much more complicated comparing to approach (1).[[BR]] 123 123 124 (A1) Overall more sophisticated than (1), can treat rapid changing temperature. [[BR]] 125 (A2) Method can be extended to treating viscosity. [[BR]] 126 (A3) Using a curl field as the source thus preventing unwanted field divergence from generating. [[BR]] 127 (D1) Computationally much more intense while still requires subcycling. The CFL is instead given by [[latex($\eta dt/l^2$)]][[BR]][[BR]] 124 128 125 129 = ====================== = … … 228 232 = ======================= = 229 233 [[BR]][[BR]] 234 (1) Flux Explusion Problem (Analytic 2D, resistivity only) [[BR]] 235 [[Image(http://www.pas.rochester.edu/~shuleli/rmhdtest1.png, 50%)]][[BR]][[BR]] 236 This problem studies the field redistribution when there is a rotating rigid cylinder with finite conductivity is placed in the flow. Can be found in text books. 237 238 (2) Hartman Flow (Analytic 2D, involves viscosity and resistivity) [[BR]] 239 [[Image(http://www.pas.rochester.edu/~shuleli/rmhdtest2.png, 50%)]][[BR]][[BR]] 240 This problem solves the state of a stationary flow of viscose and resistive fluid between two plates with superimposed transverse magnetic field. Can be found in many MHD text books and papers. 241 See for instance: 242 [[https://www.pas.rochester.edu/~shuleli/hartmann_apj08.pdf]][[BR]][[BR]] 243 244 245 (3) Hydromagnetic Rayleigh-Bernard Convection (2D or 3D. Involves thermal diffusion, viscosity, resistivity) [[BR]] 246 [[Image(http://www.pas.rochester.edu/~shuleli/rmhdtest3.png, 50%)]][[BR]][[BR]] 247 The stabilizing effect of vertical magnetic field on a classical Rayleigh-Bernard convection problem. This problem involves find the critical Chandrasekhar number with a given Rayleigh number. For a thermal convecting flow at fixed Rayleigh number, the applied vertical non-dimensional magnetic field exceeds the critical Chandrasekhar number. For details, see for instance, Chandrasekhar's 1961 book. 248 Also experiments. see for instance: 249 [[https://www.pas.rochester.edu/~shuleli/RB_pre00.pdf]][[BR]][[BR]] 250 230 251 The following LLNL journal contains useful tests on resistive MHD solver: [[BR]][[BR]] 231 252 [[https://e-reports-ext.llnl.gov/pdf/374198.pdf]][[BR]][[BR]]
