4.3 エネルギー保存則

$$v_{i,j+1}^2 + w_{i,j+1}^2 = \frac{1}{4} \left\{ \left[ (v_{i+1,j}+v_{i-1,j})^2 +2\lambda(v_{i... ..._{i-1,j})(w_{i+1,j}-w_{i-1,j}) +\lambda^2(w_{i+1,j}-w_{i-1,j})^2 \right]\right.$$

$$\left. + \left[ \lambda^2(v_{i+1,j}-v_{i-1,j})^2 +2\lambda(v_{i+1,j}-v_{i-1,j})(w_{i+1,j}+w_{i-1,j}) +(w_{i+1,j}+w_{i-1,j})^2 \right]\right\}$$

$$= \frac{1}{4}\left\{ (1+\lambda^2)(v_{i+1,j}^2+v_{i-1,j}^2) +2(1-\lambda^2)v_{i+1,j}v_{i-1,j}\right.$$

$$+ (1+\lambda^2)(w_{i+1,j}^2+w_{i-1,j}^2) +2(1-\lambda^2)w_{i+1,j}w_{i-1,j}$$

$$\left.+4\lambda(v_{i+1,j}w_{i+1,j}-v_{i-1,j}w_{i-1,j}) \right\}$$

$$= \frac{1}{4}\left\{ (1+\lambda^2)(v_{i+1,j}^2+v_{i-1,j}^2+w_{i+1,j}^2+w_{i-1,j}^2) +2(1-\lambda^2)(v_{i+1,j}v_{i-1,j}+w_{i+1,j}w_{i-1,j})\right.$$

$$\left.+4\lambda(v_{i+1,j}w_{i+1,j}-v_{i-1,j}w_{i-1,j}) \right\}.$$

$$4\sum_{i=1}^{N-1}\left(v_{i,j+1}^2+w_{i,j+1}^2\right) = (1+\lambda^2) \left(v_{0,j}^2+v_{1,j}^2+v_{N-1,j}^2+v_{N,j}^2 +w_{0,j}^2+w_{1,j}^2+w_{N-1,j}^2+w_{N,j}^2\right)$$

$$+2(1+\lambda^2)\sum_{i=2}^{N-2}(v_{i,j}^2+w_{i,j}^2)$$

$$+2(1-\lambda^2)\sum_{i=1}^{N-1} (v_{i+1,j}v_{i-1,j}+w_{i+1,j}w_{i-1,j})$$

$$+4\lambda (v_{N,j}w_{N,j}+v_{N-1,j}w_{N-1,j} -v_{1,j}w_{1,j}-v_{0,j}w_{0,j})$$

特に $\lambda =1$ の場合

$$\sum_{i=1}^{N-1}\left(v_{i,j+1}^2+w_{i,j+1}^2\right) = \frac{1}{2} \left(v_{0,j}^2+v_{1,j}^2+v_{N-1,j}^2+v_{N,j}^2 +w_{0,j}^2+w_{1,j}^2+w_{N-1,j}^2+w_{N,j}^2\right)$$

$$+\sum_{i=2}^{N-2}(v_{i,j}^2+w_{i,j}^2)$$

$$+(v_{N,j}w_{N,j}+v_{N-1,j}w_{N-1,j} -v_{1,j}w_{1,j}-v_{0,j}w_{0,j})$$

$$= \sum_{i=0}^{N}(v_{i,j}^2+w_{i,j}^2)$$

$$-\frac{1}{2} \left(v_{0,j}^2+v_{1,j}^2+v_{N-1,j}^2+v_{N,j}^2 +w_{0,j}^2+w_{1,j}^2+w_{N-1,j}^2+w_{N,j}^2\right)$$

$$-(v_{1,j}w_{1,j}+v_{0,j}w_{0,j}-v_{N,j}w_{N,j}-v_{N-1,j}w_{N-1,j})$$

$$= \sum_{i=0}^{N}(v_{i,j}^2+w_{i,j}^2)$$

$$-\frac{1}{2} \left[(v_{0,j}+w_{0,j})^2+(v_{1,j}+w_{1,j})^2 +(v_{N-1,j}-w_{N-1,j})^2+(v_{N,j}-w_{N,j})^2 \right]$$

それゆえ

$$v_{0,j+1}^2+w_{0,j+1}^2+v_{N,j+1}^2+w_{N,j+1}^2= \frac{1}{2} \lef... ...j})^2+(v_{1,j}+w_{1,j})^2 +(v_{N-1,j}-w_{N-1,j})^2+(v_{N,j}-w_{N,j})^2 \right]$$

であればエネルギー保存則が成り立つ。

簡単のため同次 Dirichlet 境界条件の場合

$$w_{0,j+1}^2+w_{N,j+1}^2= \frac{1}{2} \left[w_{0,j}^2+(v_{1,j}+w_{1,j})^2 +(v_{N-1,j}-w_{N-1,j})^2+w_{N,j}^2 \right]$$

$$w_{0,j+1}^2 =\frac{1}{2} \left[w_{0,j}^2+(v_{1,j}+w_{1,j})^2\right],\quad w_{N,j+1}^2= \frac{1}{2} \left[(v_{N-1,j}-w_{N-1,j})^2+w_{N,j}^2 \right]$$