时城有限差分法 维FDTD
1 一维FDTD 时域有限差分法
2.2一维标量波动方程的FDTD 自由空间中的 Maxwell方程 aE -VXH at ah V×E at 简单的一维情况 aE 1 aHy aH 1 dE uo az 2021/12/10
2 2.2 一维标量波动方程的FDTD 0 1 t = E H 0 1 t = − H E 自由空间中的Maxwell 方程 简单的一维情况 0 x 1 H y t z = − E 0 y 1 x H E t z = − 2021/12/10
对时间和空间差分 aE 1H, at az E12(k)-E12(k)1H(k+1/2)-H”(k-1/2) △t aH I dE at az Hy(k+1/2)-H”(k+1/2)1Ex2(k+1)-E2(k 2021/12/10
3 对时间和空间差分 0 x 1 H y t z = − E 0 y 1 x H E t z = − ( ) ( ) ( ) ( ) 1/2 1/2 0 1 1/ 2 1/ 2 n n n n x x y y k k k k t x + − − + − − = − E E H H ( ) ( ) ( ) ( ) 1 1/2 1/2 0 1/ 2 1/ 2 1 1 n n n n y y x x k k k k t x + + + + − + + − = − H H E E 2021/12/10
Ex+2(k)-E(k)1H(k+1/2)-H”(k-1/2) △t Eo △x Hn(k+1/2)-H"(k+1/2 1Ex2(k+1)-Ex2(k) LX E -12 k-2 k-1 k+2 k-11/2k-1/2 k+l/2 k+11/2 k+2l/2 E n+In k-2 k k+1 k+2 Figure 1.1 Interleaving of the E and H fields in space and time in the FDTD formulation To calculate H,(k+ 1/ 2), for instance, the neighboring values of E, at k and k+I are nceded. Similarly, to calculate Er(k+ 1). the value of H, at k+1/2 and k+I 1/2 are needed
( ) ( ) ( ) ( ) 1/2 1/2 0 1 1/ 2 1/ 2 n n n n x x y y k k k k t x + − − + − − = − E E H H ( ) ( ) ( ) ( ) 1 1/2 1/2 0 1/ 2 1/ 2 1 1 n n n n y y x x k k k k t x + + + + − + + − = − H H E E
The figure below represents the FDTD method. The orange triangles represent approximations of Er while the purple triangles represent Hr ap proximations. Each row corresponds to a specific instant in time, at half time steps, whereas each column represents a single spacial grid point through time. The gray and blue grid line represent whole and half steps respec tively, in both time and space. The initial values that must be given are the green circles, and the boundaries are the orange and purple circles (H)+2 k+ (Ex)k+2 The FDTD Approximation grid n+2 72+ 7+1 + 7 kk+2 k+l kite
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