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 |
| 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]] |
| 100 | |
| 101 | [[latex($C_0 = -(k_{X0} + ... + k_{X4})$)]][[BR]] |
| 102 | [[latex($C_1 = -k_{X0}$)]][[BR]] |
| 103 | [[latex($C_2 = k_{X0}$)]][[BR]] |
| 104 | [[latex($C_3 = k_{X2}$)]][[BR]] |
| 105 | [[latex($C_4 = -k_{X2}$)]][[BR]] |
| 106 | [[latex($C_5 = k_{X3}$)]][[BR]] |
| 107 | [[latex($C_6 = -k_{X3}$)]][[BR]] |
| 108 | [[latex($C_7 = -k_{X4}$)]][[BR]] |
| 109 | [[latex($C_8 = k_{X4}$)]][[BR]] |
| 110 | [[latex($C_9 = k_{X1}$)]][[BR]] |
| 111 | [[latex($C_{10} = k_{X2}$)]][[BR]] |
| 112 | [[latex($C_{11} = k_{X3}$)]][[BR]] |
| 113 | [[latex($C_{12} = k_{X4}$)]][[BR]] |
| 114 | |
| 115 | where k are the CFL number at the edge centers. Since the CFL are defined at the cell centers initially, we need to interpolate the edge centered CFL using the four nearest cell center temperatures. The relation between temperature and the diffusivity is described in the next section. Remember that because the x, y, z field components now reside on different face centers, the value of C vector for a certain cell will be different for different field components. [[BR]] |
| 116 | |
| 117 | The matrix equation has the form of: [[BR]] |
| 118 | |
| 119 | [[matrix2]] |
| 120 | |
| 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 | 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]] |