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Notas de Yuri.Rodrix

Página tipo blog en el que voy a publicar mis notas de aprendizaje, en especial de temas como matemáticas, física y quizá algo de programación

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Ecuación de Laplace con Condiciones de Dirichlet: Discretización Finita y Resolución por Jacobi & Gauss-Seidel

Ecuación de Laplace

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∇2ϕ=0  en el dominio Ω\nabla^{2}\phi = 0 \ \text{ en el dominio } \Omega

Condicion de Frontera

ϕ=g(x,y)  en la frontera ∂Ω\phi = g(x,y) \ \text{ en la frontera } \partial \Omega

Discretización

∇2ϕ=∂2ϕ∂x2+∂2ϕ∂y2=0\nabla^{2}\phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0∂2ϕ∂x2∣i,j≈ϕi+1,j−2 ϕi,j+ϕi−1,jh2,∂2ϕ∂y2∣i,j≈ϕi,j+1−2 ϕi,j+ϕi,j−1h2.\frac{\partial^2 \phi}{\partial x^2}\bigg|_{i,j} \approx \frac{\phi_{i+1,j} - 2\,\phi_{i,j} + \phi_{i-1,j}}{h^2}, \qquad \frac{\partial^2 \phi}{\partial y^2}\bigg|_{i,j} \approx \frac{\phi_{i,j+1} - 2\,\phi_{i,j} + \phi_{i,j-1}}{h^2}.ϕi+1,j−2 ϕi,j+ϕi−1,jh2+ϕi,j+1−2 ϕi,j+ϕi,j−1h2=0\frac{\phi_{i+1,j} - 2\,\phi_{i,j} + \phi_{i-1,j}}{h^2} + \frac{\phi_{i,j+1} - 2\,\phi_{i,j} + \phi_{i,j-1}}{h^2} = 0ϕi+1,j−2 ϕi,j+ϕi−1,j+ϕi,j+1−2 ϕi,j+ϕi,j−1=0\phi_{i+1,j} - 2\,\phi_{i,j} + \phi_{i-1,j} + \phi_{i,j+1} - 2\,\phi_{i,j} + \phi_{i,j-1} = 0−4 ϕi,j+ϕi+1,j+ϕi−1,j+ϕi,j+1+ϕi,j−1=0- 4\,\phi_{i,j} + \phi_{i+1,j} +\phi_{i-1,j} + \phi_{i,j+1} + \phi_{i,j-1} = 0ϕi+1,j+ϕi−1,j+ϕi,j+1+ϕi,j−1=4 ϕi,j\phi_{i+1,j} +\phi_{i-1,j} + \phi_{i,j+1} + \phi_{i,j-1} = 4\,\phi_{i,j}14(ϕi+1,j+ϕi−1,j+ϕi,j+1+ϕi,j−1)=ϕi,j\frac{1}{4}( \phi_{i+1,j} +\phi_{i-1,j} + \phi_{i,j+1} + \phi_{i,j-1})= \phi_{i,j}

Método de Jacobi

(D+L+U)x=b(D+L+U)x=bDx+(L+U)x=bDx+(L+U)x=bDx=−(L+U)x+bDx=-(L+U)x+bDx(k+1)=−(L+U)x(k)+bDx^{(k+1)}=-(L+U)x^{(k)}+bx=−D−1(L+U)⏟Tjx+D−1⏟Cjbx=\underset{T_j}{\underbrace{-D^{-1}(L+U)}}x+\underset{C_j}{\underbrace{D^{-1}}}b
xi(k+1)  =  1aii(bi  −  ∑j=1j≠inaij xj(k)).x_i^{(k+1)} \;=\; \frac{1}{a_{ii}} \Bigl( b_i \;-\; \sum_{\substack{j=1 \\ j\neq i}}^{n} a_{ij}\,x_j^{(k)} \Bigr).

METJacobiCode
Convergencia
ρ(Tj)<1\Large \rho(T_j)<1

M. de Gauss-Seidel

(D+L+U)x=b(D+L+U)x=b(D+L)x+Ux=b(D+L)x +Ux=b(D+L)x=−Ux+b(D+L)x=-Ux+b(D+L)x(k+1)=−Ux(k)+b(D+L)x^{(k+1)}=-Ux^{(k)}+bx=−(D+L)−1U⏟Tgx+(D+L)−1⏟Cgbx=\underset{T_g}{\underbrace{-(D+L)^{-1}U}}x+\underset{C_g}{\underbrace{(D+L)^{-1}}}b
xi(k+1)  =  1aii(bi  −  ∑j=1i−1aij xj(k+1)  −  ∑j=i+1naij xj(k)).x_i^{(k+1)} \;=\; \frac{1}{a_{ii}} \Bigl( b_i \;-\; \sum_{j=1}^{i-1} a_{ij}\,x_j^{(k+1)} \;-\; \sum_{j=i+1}^{n} a_{ij}\,x_j^{(k)} \Bigr).

METGaussSeidelCode
Convergencia
ρ(Tg)<1\Large \rho(T_g)<1


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