Mudanças entre as edições de "Análise espectral e estabilidade"
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A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo. | A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo. | ||
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| + | <math>\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}= f(u)\,\!</math> | ||
==O problema== | ==O problema== | ||
Edição das 14h40min de 10 de maio de 2010
A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo.
<math>\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}= f(u)\,\!</math>
Índice
O problema
Queremos aproximar a solução da equação de Navier Stokes em um duto. Para isso devemos resolver a cada passo de tempo uma equação de Poisson como <math>\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}= f(u)\,\!</math>
que é discretizada como
<math>a p_{i+1,j}+a p_{i-1,j}+c p_{i,j} +b p_{i,j+1}+b p_{i,j-1} = f_{i,j} \,\!</math>
- A matriz possui o estêncil [...b,0...0,a,c,a,0...0,b,0...] onde
<math>a=\frac{1}{dx^2}, b=\frac{1}{dy^2}, c=-2a-2b \,\!</math>
- O espectro do problema não depende de dt, Re ou U0. Depende apenas de a, b e c.
- Para montar a matriz de iteração corretamente usamos como condição de Dirichlet
p(contorno) = p0
- e dentro do loop de iteração temos
pnew(contorno)= p(contorno)
- ou seja, p continua sempre fixo.
Método Iterativo
Para solucionar o sistema
- <math>M p = f</math>
usaremos um método iterativo da forma
- <math> p^{(k+1)} = G p^{(k)}+g </math>
Para que esse método seja convergente é necessário que o raio espectral de G seja estritamente menor do que 1,
- <math> \rho(G) < 1</math>
Método de Jacobi
Tomando
- <math>M=D+L+U</math>
o sistema pode ser reescrito como
- <math>D p = -(L+U)p +f </math>
- <math> p^{(k+1)} = - D^{-1}(L+U) p^{(k)}+D^{-1}f </math>
- <math> p^{(k+1)} = G p^{(k)}+D^{-1}f </math>
onde
- <math> G = - D^{-1}(L+U)</math>
Método de Gauss-Seidel
Tomando
- <math>M=D+L+U</math>
o sistema pode ser reescrito como
- <math>(D+L)p = -Up +f </math>
- <math> p^{(k+1)} = - (D+L)^{-1}U p^{(k)}+(D+L)^{-1}f </math>
- <math> p^{(k+1)} = G p^{(k)}+(D+L)^{-1}f </math>
onde
- <math> G_{GS} = - (D+L)^{-1}U</math>
Método SOR
Usando o método de Gauss-Seidel como base, temos
- <math> p^{(k+1)} =(1-w)p^{(k)}+w( G p^{(k)}+(D+L)^{-1}f) </math>
onde
- <math> G_{SOR} = (1-w)I+wG = I + w(G-I)</math>
Comparação
Comparação do número de iterações para a convergência dos métodos de Jacobi, Gauss-Seidel e SOR (com ω=1.7) para a equação de Poisson com [m, n]=[11, 11]
- Configuração A
- Condição de contorno de Neumann
- pnew=pnew-p(1,1)
- |λ[1]|<1
- int(f,dv) não é zero (não é divergente free)
- Configuração B
- Condição de contorno de Neumann
- |λ[1]|=1
- int(f,dv)=zero (é divergente free)
Os gráficos abaixo são para a configuração A, porém com a configuração B temos o mesmo comportamento.
Condição de Dirichlet em todos os lados
Antes de entrar no loop fixamos
p( :, 1)= p_sul p( :,ny)= p_norte p( 1, :)= p_oeste p(nx, :)= p_leste
Para obter a matriz de iteração foi necessário usar as CC.
pnew( :, 1)= p( :, 1) pnew( :,ny)= p( :,ny) pnew( 1, :)= p( 1, :) pnew(nx, :)= p(nx, :)
O espectro é real com 0<|λ|<1. A condição de contorno contribui com autovalores λ=1.
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830
lambda(n)= 4.218228659668808E-003 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
(-0.955 , 0 ) (-0.955 , 0 )
( 0.955 , 0 ) ( 0.955 , 0 )
( 0.896 , 0 ) (-0.896 , 0 )
(-0.896 , 0 ) ( 0.896 , 0 )
(-0.884 , 0 ) ( 0.884 , 0 )
( 0.884 , 0 ) (-0.884 , 0 )
( 0.825 , 0 ) ( 0.825 , 0 )
(-0.825 , 0 ) (-0.825 , 0 )
( 0.803 , 0 ) (-0.803 , 0 )
(-0.803 , 0 ) ( 0.803 , 0 )
(-0.774 , 0 ) ( 0.774 , 0 )
( 0.774 , 0 ) (-0.774 , 0 )
( 0.732 , 0 ) ( 0.732 , 0 )
(-0.732 , 0 ) (-0.732 , 0 )
(-0.715 , 0 ) ( 0.715 , 0 )
( 0.715 , 0 ) (-0.715 , 0 )
( 0.683 , 0 ) ( 0.683 , 0 )
(-0.683 , 0 ) (-0.683 , 0 )
(-0.634 , 0 ) (-0.634 , 0 )
( 0.634 , 0 ) ( 0.634 , 0 )
( 0.621 , 0 ) ( 0.621 , 0 )
(-0.621 , 0 ) (-0.621 , 0 )
(-0.612 , 0 ) ( 0.612 , 0 )
( 0.612 , 0 ) (-0.612 , 0 )
( 0.575 , 0 ) ( 0.575 , 0 )
(-0.575 , 0 ) (-0.575 , 0 )
(-0.547 , 0 ) ( 0.547 , 0 )
( 0.547 , 0 ) (-0.547 , 0 )
( 0.502 , 0 ) ( 0.502 , 0 )
(-0.502 , 0 ) (-0.502 , 0 )
(-0.482 , 0 ) ( 0.482 , 0 )
( 0.482 , 0 ) (-0.482 , 0 )
(-0.480 , 0 ) ( 0.480 , 0 )
( 0.480 , 0 ) (-0.480 , 0 )
(-0.476 , 0 ) ( 0.476 , 0 )
( 0.476 , 0 ) (-0.476 , 0 )
( 0.421 , 0 ) ( 0.421 , 0 )
(-0.421 , 0 ) (-0.421 , 0 )
(-0.404 , 0 ) ( 0.404 , 0 )
( 0.404 , 0 ) (-0.404 , 0 )
(-0.365 , 0 ) ( 0.365 , 0 )
( 0.365 , 0 ) (-0.365 , 0 )
( 0.362 , 0 ) ( 0.362 , 0 )
(-0.362 , 0 ) (-0.362 , 0 )
( 0.333 , 0 ) (-0.333 , 0 )
(-0.333 , 0 ) ( 0.333 , 0 )
(-0.327 , 0 ) ( 0.327 , 0 )
( 0.327 , 0 ) (-0.327 , 0 )
( 0.325 , 0 ) (-0.325 , 0 )
(-0.325 , 0 ) ( 0.325 , 0 )
( 0.268 , 0 ) (-0.268 , 0 )
(-0.268 , 0 ) ( 0.268 , 0 )
( 0.266 , 0 ) (-0.266 , 0 )
(-0.266 , 0 ) ( 0.266 , 0 )
( 0.226 , 0 ) ( 0.226 , 0 )
(-0.226 , 0 ) (-0.226 , 0 )
(-0.223 , 0 ) ( 0.223 , 0 )
( 0.223 , 0 ) (-0.223 , 0 )
(-0.208 , 0 ) (-0.208 , 0 )
( 0.208 , 0 ) ( 0.208 , 0 )
(-0.197 , 0 ) ( 0.197 , 0 )
( 0.197 , 0 ) (-0.197 , 0 )
(-0.186 , 0 ) ( 0.186 , 0 )
( 0.186 , 0 ) (-0.186 , 0 )
( 0.173 , 0 ) ( 0.173 , 0 )
(-0.173 , 0 ) (-0.173 , 0 )
( 0.148 , 0 ) ( 0.148 , 0 )
(-0.148 , 0 ) (-0.148 , 0 )
(-0.127 , 0 ) ( 0.127 , 0 )
( 0.127 , 0 ) (-0.127 , 0 )
(-0.862E-01, 0 ) (-0.862E-01, 0 )
( 0.862E-01, 0 ) ( 0.862E-01, 0 )
(-0.834E-01, 0 ) (-0.834E-01, 0 )
( 0.834E-01, 0 ) ( 0.834E-01, 0 )
( 0.771E-01, 0 ) (-0.771E-01, 0 )
(-0.771E-01, 0 ) ( 0.771E-01, 0 )
( 0.752E-01, 0 ) (-0.752E-01, 0 )
(-0.752E-01, 0 ) ( 0.752E-01, 0 )
(-0.712E-01, 0 ) (-0.712E-01, 0 )
( 0.712E-01, 0 ) ( 0.712E-01, 0 )
(-0.549E-01, 0 ) ( 0.549E-01, 0 )
( 0.549E-01, 0 ) (-0.549E-01, 0 )
( 0.532E-01, 0 ) (-0.532E-01, 0 )
(-0.532E-01, 0 ) ( 0.532E-01, 0 )
(-0.335E-01, 0 ) ( 0.335E-01, 0 )
( 0.335E-01, 0 ) (-0.335E-01, 0 )
(-0.161E-01, 0 ) ( 0.161E-01, 0 )
( 0.161E-01, 0 ) (-0.161E-01, 0 )
( 0.422E-02, 0 ) ( 0.422E-02, 0 )
(-0.422E-02, 0 ) (-0.422E-02, 0 )
Condição de Neumann em todos os lados
Usando as CC
pnew( :,1 ) = pnew(:,2) pnew( :,ny) = pnew(:,ny-1) pnew( 1,: ) = pnew(2,:) pnew(nx,: ) = pnew(nx-1,:)
O espectro é real com σ⊂(-1,1).
Incluindo CC na matriz Eliminando CC na matriz
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830
lambda(n)= 0.000000000000000 0 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 1 , 0 ) (-0.955 , 0 )
( 0.976 , 0 ) ( 0.955 , 0 )
( 0.970 , 0 ) (-0.896 , 0 )
(-0.945 , 0 ) ( 0.896 , 0 )
( 0.945 , 0 ) ( 0.884 , 0 )
( 0.905 , 0 ) (-0.884 , 0 )
( 0.883 , 0 ) ( 0.825 , 0 )
(-0.874 , 0 ) (-0.825 , 0 )
( 0.874 , 0 ) (-0.803 , 0 )
( 0.859 , 0 ) ( 0.803 , 0 )
(-0.859 , 0 ) ( 0.774 , 0 )
( 0.794 , 0 ) (-0.774 , 0 )
(-0.788 , 0 ) ( 0.732 , 0 )
( 0.788 , 0 ) (-0.732 , 0 )
( 0.764 , 0 ) ( 0.715 , 0 )
(-0.764 , 0 ) (-0.715 , 0 )
( 0.750 , 0 ) ( 0.683 , 0 )
(-0.726 , 0 ) (-0.683 , 0 )
( 0.726 , 0 ) (-0.634 , 0 )
( 0.677 , 0 ) ( 0.634 , 0 )
(-0.677 , 0 ) ( 0.621 , 0 )
(-0.655 , 0 ) (-0.621 , 0 )
( 0.655 , 0 ) ( 0.612 , 0 )
( 0.655 , 0 ) (-0.612 , 0 )
( 0.624 , 0 ) ( 0.575 , 0 )
(-0.624 , 0 ) (-0.575 , 0 )
( 0.587 , 0 ) ( 0.547 , 0 )
(-0.562 , 0 ) (-0.547 , 0 )
( 0.562 , 0 ) ( 0.502 , 0 )
(-0.544 , 0 ) (-0.502 , 0 )
( 0.544 , 0 ) ( 0.482 , 0 )
( 0.538 , 0 ) (-0.482 , 0 )
(-0.538 , 0 ) ( 0.480 , 0 )
( 0.500 , 0 ) (-0.480 , 0 )
( 0.491 , 0 ) ( 0.476 , 0 )
(-0.491 , 0 ) (-0.476 , 0 )
(-0.470 , 0 ) ( 0.421 , 0 )
( 0.470 , 0 ) (-0.421 , 0 )
( 0.413 , 0 ) ( 0.404 , 0 )
(-0.405 , 0 ) (-0.404 , 0 )
( 0.405 , 0 ) ( 0.365 , 0 )
(-0.389 , 0 ) (-0.365 , 0 )
( 0.389 , 0 ) ( 0.362 , 0 )
(-0.383 , 0 ) (-0.362 , 0 )
( 0.383 , 0 ) (-0.333 , 0 )
(-0.381 , 0 ) ( 0.333 , 0 )
( 0.381 , 0 ) ( 0.327 , 0 )
( 0.345 , 0 ) (-0.327 , 0 )
(-0.318 , 0 ) (-0.325 , 0 )
( 0.318 , 0 ) ( 0.325 , 0 )
( 0.315 , 0 ) (-0.268 , 0 )
(-0.315 , 0 ) ( 0.268 , 0 )
(-0.250 , 0 ) (-0.266 , 0 )
( 0.250 , 0 ) ( 0.266 , 0 )
( 0.250 , 0 ) ( 0.226 , 0 )
(-0.241 , 0 ) (-0.226 , 0 )
( 0.241 , 0 ) ( 0.223 , 0 )
(-0.229 , 0 ) (-0.223 , 0 )
( 0.229 , 0 ) (-0.208 , 0 )
(-0.226 , 0 ) ( 0.208 , 0 )
( 0.226 , 0 ) ( 0.197 , 0 )
( 0.207 , 0 ) (-0.197 , 0 )
(-0.207 , 0 ) ( 0.186 , 0 )
( 0.206 , 0 ) (-0.186 , 0 )
( 0.176 , 0 ) ( 0.173 , 0 )
(-0.176 , 0 ) (-0.173 , 0 )
(-0.155 , 0 ) ( 0.148 , 0 )
( 0.155 , 0 ) (-0.148 , 0 )
( 0.117 , 0 ) ( 0.127 , 0 )
( 0.955E-01, 0 ) (-0.127 , 0 )
( 0.955E-01, 0 ) (-0.862E-01, 0 )
(-0.955E-01, 0 ) ( 0.862E-01, 0 )
(-0.925E-01, 0 ) (-0.834E-01, 0 )
( 0.925E-01, 0 ) ( 0.834E-01, 0 )
(-0.891E-01, 0 ) (-0.771E-01, 0 )
( 0.891E-01, 0 ) ( 0.771E-01, 0 )
(-0.868E-01, 0 ) (-0.752E-01, 0 )
( 0.868E-01, 0 ) ( 0.752E-01, 0 )
( 0.677E-01, 0 ) (-0.712E-01, 0 )
(-0.677E-01, 0 ) ( 0.712E-01, 0 )
(-0.653E-01, 0 ) ( 0.549E-01, 0 )
( 0.653E-01, 0 ) (-0.549E-01, 0 )
( 0.439E-01, 0 ) (-0.532E-01, 0 )
(-0.439E-01, 0 ) ( 0.532E-01, 0 )
( 0.302E-01, 0 ) ( 0.335E-01, 0 )
( 0.245E-01, 0 ) (-0.335E-01, 0 )
( 0.215E-01, 0 ) ( 0.161E-01, 0 )
(-0.215E-01, 0 ) (-0.161E-01, 0 )
(-0.568E-02, 0 ) ( 0.422E-02, 0 )
( 0.568E-02, 0 ) (-0.422E-02, 0 )
( 0.251E-14, 0.857E-15)
( 0.251E-14,-0.857E-15)
(-0.255E-14, 0 )
( 0.233E-15, 0.235E-14)
( 0.233E-15,-0.235E-14)
(-0.124E-14, 0.565E-15)
(-0.124E-14,-0.565E-15)
( 0.113E-14, 0 )
(-0.772E-15, 0 )
( 0.632E-15, 0.207E-15)
( 0.632E-15,-0.207E-15)
(-0.490E-15, 0.439E-15)
(-0.490E-15,-0.439E-15)
( 0.165E-15, 0.488E-15)
( 0.165E-15,-0.488E-15)
( 0.458E-15, 0 )
(-0.433E-15, 0 )
(-0.330E-15, 0.101E-15)
(-0.330E-15,-0.101E-15)
( 0.212E-15, 0.141E-15)
( 0.212E-15,-0.141E-15)
(-0.123E-15, 0.217E-15)
(-0.123E-15,-0.217E-15)
( 0.750E-17, 0.128E-15)
( 0.750E-17,-0.128E-15)
( 0.277E-16, 0.412E-16)
( 0.277E-16,-0.412E-16)
(-0.217E-16, 0.252E-16)
(-0.217E-16,-0.252E-16)
( 0.327E-16, 0 )
(-0.230E-16, 0 )
(-0.824E-30, 0.885E-30)
(-0.824E-30,-0.885E-30)
( 0.680E-30, 0 )
(-0.684E-31, 0 )
( 0.317E-31, 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
Condição de Neumann em todos os lados e subtraindo P(1,2)
Usando
Pnew=Pnew -P(1,2)
temos o espectro abaixo.
Incluindo CC na Matriz Excluindo CC na Matriz lambda(1)= 0.975528258147575 lambda(1)= 0.955274744954828 lambda(n)= 0.000000000000000 0 lambda(n)= 0.000000000000000 0
( 0.976 , 0.0 ) (-0.955 , 0.0 ) ( 0.970 , 0.0 ) ( 0.911 , 0.0 ) ( 0.945 , 0.0 ) ( 0.896 , 0.0 ) (-0.945 , 0.0 ) (-0.896 , 0.0 ) ( 0.905 , 0.0 ) ( 0.884 , 0.0 ) ( 0.883 , 0.0 ) (-0.884 , 0.0 ) (-0.874 , 0.0 ) (-0.825 , 0.0 ) ( 0.874 , 0.0 ) ( 0.825 , 0.0 ) ( 0.859 , 0.0 ) (-0.803 , 0.0 ) (-0.859 , 0.0 ) ( 0.789 , 0.0 ) ( 0.794 , 0.0 ) (-0.774 , 0.0 ) ( 0.788 , 0.0 ) (-0.732 , 0.0 ) (-0.788 , 0.0 ) ( 0.732 , 0.0 ) ( 0.764 , 0.0 ) ( 0.715 , 0.0 ) (-0.764 , 0.0 ) (-0.712 , 0.0 ) ( 0.750 , 0.0 ) ( 0.693 , 0.0 ) (-0.726 , 0.0 ) ( 0.683 , 0.0 ) ( 0.726 , 0.0 ) (-0.682 , 0.0 ) (-0.677 , 0.0 ) (-0.634 , 0.0 ) ( 0.677 , 0.0 ) ( 0.634 , 0.0 ) ( 0.655 , 0.0 ) (-0.621 , 0.0 ) (-0.655 , 0.0 ) (-0.612 , 0.0 ) ( 0.655 , 0.0 ) ( 0.612 , 0.0 ) (-0.624 , 0.0 ) ( 0.583 , 0.0 ) ( 0.624 , 0.0 ) (-0.575 , 0.0 ) ( 0.587 , 0.0 ) ( 0.575 , 0.0 ) ( 0.562 , 0.0 ) (-0.547 , 0.0 ) (-0.562 , 0.0 ) ( 0.507 , 0.0 ) ( 0.544 , 0.0 ) ( 0.502 , 0.0 ) (-0.544 , 0.0 ) (-0.494 , 0.0 ) (-0.538 , 0.0 ) ( 0.482 , 0.0 ) ( 0.538 , 0.0 ) (-0.482 , 0.0 ) ( 0.500 , 0.0 ) (-0.480 , 0.0 ) ( 0.491 , 0.0 ) (-0.476 , 0.0 ) (-0.491 , 0.0 ) ( 0.476 , 0.0 ) (-0.470 , 0.0 ) ( 0.421 , 0.0 ) ( 0.470 , 0.0 ) (-0.416 , 0.0 ) ( 0.413 , 0.0 ) ( 0.412 , 0.0 ) (-0.405 , 0.0 ) ( 0.404 , 0.0 ) ( 0.405 , 0.0 ) (-0.402 , 0.0 ) (-0.389 , 0.0 ) (-0.365 , 0.0 ) ( 0.389 , 0.0 ) (-0.362 , 0.0 ) ( 0.383 , 0.0 ) ( 0.362 , 0.0 ) (-0.383 , 0.0 ) ( 0.343 , 0.0 ) (-0.381 , 0.0 ) ( 0.333 , 0.0 ) ( 0.381 , 0.0 ) (-0.333 , 0.0 ) ( 0.345 , 0.0 ) (-0.327 , 0.0 ) ( 0.318 , 0.0 ) (-0.325 , 0.0 ) (-0.318 , 0.0 ) ( 0.325 , 0.0 ) ( 0.315 , 0.0 ) ( 0.283 , 0.0 ) (-0.315 , 0.0 ) (-0.268 , 0.0 ) ( 0.250 , 0.0 ) ( 0.266 , 0.0 ) (-0.250 , 0.0 ) (-0.266 , 0.0 ) ( 0.250 , 0.0 ) (-0.226 , 0.0 ) ( 0.241 , 0.0 ) ( 0.226 , 0.0 ) (-0.241 , 0.0 ) ( 0.223 , 0.0 ) (-0.229 , 0.0 ) (-0.218 , 0.0 ) ( 0.229 , 0.0 ) ( 0.208 , 0.0 ) (-0.226 , 0.0 ) ( 0.202 , 0.0 ) ( 0.226 , 0.0 ) (-0.197 , 0.0 ) ( 0.207 , 0.0 ) ( 0.197 , 0.0 ) (-0.207 , 0.0 ) (-0.190 , 0.0 ) ( 0.206 , 0.0 ) (-0.186 , 0.0 ) ( 0.176 , 0.0 ) (-0.173 , 0.0 ) (-0.176 , 0.0 ) ( 0.173 , 0.0 ) ( 0.155 , 0.0 ) ( 0.148 , 0.0 ) (-0.155 , 0.0 ) (-0.146 , 0.0 ) ( 0.117 , 0.0 ) ( 0.127 , 0.0 ) (-0.955E-01, 0.0 ) ( 0.122 , 0.0 ) ( 0.955E-01, 0.411E-15) (-0.120 , 0.0 ) ( 0.955E-01,-0.411E-15) (-0.862E-01, 0.0 ) ( 0.925E-01, 0.0 ) (-0.834E-01, 0.0 ) (-0.925E-01, 0.0 ) ( 0.834E-01, 0.0 ) (-0.891E-01, 0.0 ) ( 0.789E-01, 0.0 ) ( 0.891E-01, 0.0 ) (-0.771E-01, 0.0 ) ( 0.868E-01, 0.0 ) ( 0.771E-01, 0.0 ) (-0.868E-01, 0.0 ) (-0.752E-01, 0.0 ) (-0.677E-01, 0.0 ) ( 0.752E-01, 0.0 ) ( 0.677E-01, 0.0 ) (-0.712E-01, 0.0 ) ( 0.653E-01, 0.0 ) ( 0.573E-01, 0.0 ) (-0.653E-01, 0.0 ) (-0.549E-01, 0.0 ) (-0.439E-01, 0.0 ) (-0.532E-01, 0.0 ) ( 0.439E-01, 0.0 ) ( 0.532E-01, 0.0 ) ( 0.302E-01, 0.0 ) ( 0.395E-01, 0.0 ) ( 0.245E-01, 0.0 ) (-0.335E-01, 0.0 ) (-0.215E-01, 0.0 ) (-0.161E-01, 0.0 ) ( 0.215E-01, 0.0 ) ( 0.161E-01, 0.0 ) ( 0.568E-02, 0.0 ) ( 0.523E-02, 0.0 ) (-0.568E-02, 0.0 ) (-0.422E-02, 0.0 ) (-0.444E-14, 0.0 ) ( 0.0 , 0.0 ) ( 0.214E-14, 0.128E-14) ( 0.214E-14,-0.128E-14) ( 0.217E-14, 0.0 ) (-0.168E-14, 0.0 ) (-0.140E-14, 0.798E-15) (-0.140E-14,-0.798E-15) (-0.890E-15, 0.126E-14) (-0.890E-15,-0.126E-14) ( 0.565E-15, 0.105E-14) ( 0.565E-15,-0.105E-14) ( 0.876E-15, 0.0 ) (-0.773E-15, 0.208E-15) (-0.773E-15,-0.208E-15) ( 0.587E-15, 0.0 ) (-0.150E-16, 0.567E-15) (-0.150E-16,-0.567E-15) ( 0.352E-15, 0.337E-15) ( 0.352E-15,-0.337E-15) ( 0.127E-15, 0.324E-15) ( 0.127E-15,-0.324E-15) (-0.137E-16, 0.199E-15) (-0.137E-16,-0.199E-15) ( 0.172E-15, 0.372E-16) ( 0.172E-15,-0.372E-16) ( 0.719E-16, 0.157E-15) ( 0.719E-16,-0.157E-15) (-0.153E-15, 0.661E-16) (-0.153E-15,-0.661E-16) (-0.304E-16, 0.683E-16) (-0.304E-16,-0.683E-16) (-0.192E-16, 0.0 ) (-0.955E-28, 0.0 ) (-0.281E-30, 0.325E-30) (-0.281E-30,-0.325E-30) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 )
Condição de Neumann em todos os lados fixando um ponto, P(1,2)=1
Usando CC de Neumann e fixando em um ponto
Pnew(1,2)=P(1,2)
Temos o espectro abaixo.
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830
lambda(n)= 0.000000000000000 0 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 1 , 0 ) (-0.955 , 0 )
( 0.999 , 0 ) ( 0.955 , 0 )
( 0.974 , 0 ) (-0.896 , 0 )
( 0.967 , 0 ) ( 0.896 , 0 )
(-0.945 , 0 ) ( 0.884 , 0 )
( 0.939 , 0 ) (-0.884 , 0 )
( 0.902 , 0 ) ( 0.825 , 0 )
( 0.881 , 0 ) (-0.825 , 0 )
(-0.874 , 0 ) (-0.803 , 0 )
( 0.869 , 0 ) ( 0.803 , 0 )
(-0.859 , 0 ) ( 0.774 , 0 )
( 0.849 , 0 ) (-0.774 , 0 )
( 0.793 , 0 ) ( 0.732 , 0 )
(-0.788 , 0 ) (-0.732 , 0 )
( 0.782 , 0 ) ( 0.715 , 0 )
(-0.764 , 0 ) (-0.715 , 0 )
( 0.758 , 0 ) ( 0.683 , 0 )
( 0.746 , 0 ) (-0.683 , 0 )
(-0.726 , 0 ) (-0.634 , 0 )
( 0.718 , 0 ) ( 0.634 , 0 )
(-0.677 , 0 ) ( 0.621 , 0 )
( 0.672 , 0 ) (-0.621 , 0 )
(-0.655 , 0 ) ( 0.612 , 0 )
( 0.655 , 0 ) (-0.612 , 0 )
( 0.645 , 0 ) ( 0.575 , 0 )
(-0.625 , 0 ) (-0.575 , 0 )
( 0.617 , 0 ) ( 0.547 , 0 )
( 0.584 , 0 ) (-0.547 , 0 )
(-0.563 , 0 ) ( 0.502 , 0 )
( 0.558 , 0 ) (-0.502 , 0 )
(-0.545 , 0 ) ( 0.482 , 0 )
( 0.541 , 0 ) (-0.482 , 0 )
(-0.538 , 0 ) ( 0.480 , 0 )
( 0.527 , 0 ) (-0.480 , 0 )
( 0.498 , 0 ) ( 0.476 , 0 )
(-0.492 , 0 ) (-0.476 , 0 )
( 0.485 , 0 ) ( 0.421 , 0 )
(-0.470 , 0 ) (-0.421 , 0 )
( 0.462 , 0 ) ( 0.404 , 0 )
( 0.412 , 0 ) (-0.404 , 0 )
(-0.406 , 0 ) ( 0.365 , 0 )
( 0.401 , 0 ) (-0.365 , 0 )
(-0.389 , 0 ) ( 0.362 , 0 )
( 0.387 , 0 ) (-0.362 , 0 )
(-0.384 , 0 ) (-0.333 , 0 )
( 0.382 , 0 ) ( 0.333 , 0 )
(-0.382 , 0 ) ( 0.327 , 0 )
( 0.366 , 0 ) (-0.327 , 0 )
( 0.343 , 0 ) (-0.325 , 0 )
(-0.319 , 0 ) ( 0.325 , 0 )
( 0.317 , 0 ) (-0.268 , 0 )
(-0.316 , 0 ) ( 0.268 , 0 )
( 0.306 , 0 ) (-0.266 , 0 )
(-0.253 , 0 ) ( 0.266 , 0 )
( 0.250 , 0 ) ( 0.226 , 0 )
( 0.247 , 0 ) (-0.226 , 0 )
(-0.243 , 0 ) ( 0.223 , 0 )
( 0.237 , 0 ) (-0.223 , 0 )
(-0.230 , 0 ) (-0.208 , 0 )
( 0.227 , 0 ) ( 0.208 , 0 )
(-0.226 , 0 ) ( 0.197 , 0 )
( 0.219 , 0 ) (-0.197 , 0 )
(-0.209 , 0 ) ( 0.186 , 0 )
( 0.206 , 0 ) (-0.186 , 0 )
( 0.197 , 0 ) ( 0.173 , 0 )
(-0.176 , 0 ) (-0.173 , 0 )
( 0.172 , 0 ) ( 0.148 , 0 )
(-0.156 , 0 ) (-0.148 , 0 )
( 0.150 , 0 ) ( 0.127 , 0 )
( 0.116 , 0 ) (-0.127 , 0 )
(-0.102 , 0 ) (-0.862E-01, 0 )
( 0.955E-01, 0 ) ( 0.862E-01, 0 )
( 0.941E-01, 0 ) (-0.834E-01, 0 )
(-0.930E-01, 0 ) ( 0.834E-01, 0 )
( 0.916E-01, 0 ) (-0.771E-01, 0 )
(-0.911E-01, 0 ) ( 0.771E-01, 0 )
( 0.882E-01, 0 ) (-0.752E-01, 0 )
(-0.881E-01, 0 ) ( 0.752E-01, 0 )
( 0.795E-01, 0 ) (-0.712E-01, 0 )
(-0.708E-01, 0 ) ( 0.712E-01, 0 )
( 0.660E-01, 0 ) ( 0.549E-01, 0 )
(-0.655E-01, 0 ) (-0.549E-01, 0 )
( 0.587E-01, 0 ) (-0.532E-01, 0 )
(-0.463E-01, 0 ) ( 0.532E-01, 0 )
( 0.389E-01, 0 ) ( 0.335E-01, 0 )
( 0.298E-01, 0 ) (-0.335E-01, 0 )
( 0.243E-01, 0 ) ( 0.161E-01, 0 )
(-0.229E-01, 0 ) (-0.161E-01, 0 )
( 0.186E-01, 0 ) ( 0.422E-02, 0 )
(-0.615E-02, 0 ) (-0.422E-02, 0 )
( 0.500E-02, 0 )
(-0.799E-14, 0 )
( 0.595E-14, 0 )
(-0.228E-14, 0 )
(-0.173E-14, 0.900E-15)
(-0.173E-14,-0.900E-15)
( 0.176E-14, 0 )
( 0.906E-15, 0.488E-15)
( 0.906E-15,-0.488E-15)
(-0.669E-15, 0.726E-15)
(-0.669E-15,-0.726E-15)
( 0.756E-15, 0 )
(-0.694E-15, 0 )
(-0.592E-16, 0.509E-15)
(-0.592E-16,-0.509E-15)
( 0.451E-15, 0.209E-15)
( 0.451E-15,-0.209E-15)
( 0.995E-16, 0.355E-15)
( 0.995E-16,-0.355E-15)
(-0.226E-15, 0.206E-15)
(-0.226E-15,-0.206E-15)
(-0.240E-15, 0 )
( 0.157E-15, 0 )
( 0.110E-16, 0.108E-15)
( 0.110E-16,-0.108E-15)
(-0.407E-16, 0.954E-16)
(-0.407E-16,-0.954E-16)
( 0.798E-16, 0.494E-16)
( 0.798E-16,-0.494E-16)
(-0.372E-16, 0 )
( 0.172E-17, 0.943E-17)
( 0.172E-17,-0.943E-17)
( 0.454E-26, 0 )
(-0.800E-30, 0.213E-30)
(-0.800E-30,-0.213E-30)
( 0.683E-30, 0 )
(-0.556E-31, 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
Condição de Neumann em 3 lados fixando P=1 na entrada (ou na saída)
Fixando a entrada oeste com
Pnew(1,:)=P(1,:)
e o resto com CC de Neumann temos o espectro
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830
lambda(n)= 0.000000000000000 0 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 1 , 0 )
( 0.994 , 0 ) (-0.955 , 0 )
( 0.964 , 0 ) ( 0.955 , 0 )
( 0.950 , 0 ) (-0.896 , 0 )
(-0.948 , 0 ) ( 0.896 , 0 )
( 0.920 , 0 ) ( 0.884 , 0 )
(-0.883 , 0 ) (-0.884 , 0 )
( 0.877 , 0 ) ( 0.825 , 0 )
( 0.867 , 0 ) (-0.825 , 0 )
(-0.861 , 0 ) (-0.803 , 0 )
( 0.836 , 0 ) ( 0.803 , 0 )
( 0.834 , 0 ) ( 0.774 , 0 )
(-0.796 , 0 ) (-0.774 , 0 )
(-0.782 , 0 ) ( 0.732 , 0 )
( 0.750 , 0 ) (-0.732 , 0 )
( 0.750 , 0 ) ( 0.715 , 0 )
( 0.744 , 0 ) (-0.715 , 0 )
(-0.728 , 0 ) ( 0.683 , 0 )
( 0.720 , 0 ) (-0.683 , 0 )
( 0.700 , 0 ) (-0.634 , 0 )
(-0.695 , 0 ) ( 0.634 , 0 )
(-0.663 , 0 ) ( 0.621 , 0 )
(-0.653 , 0 ) (-0.621 , 0 )
( 0.633 , 0 ) ( 0.612 , 0 )
( 0.617 , 0 ) (-0.612 , 0 )
( 0.611 , 0 ) ( 0.575 , 0 )
( 0.581 , 0 ) (-0.575 , 0 )
( 0.581 , 0 ) ( 0.547 , 0 )
(-0.566 , 0 ) (-0.547 , 0 )
(-0.565 , 0 ) ( 0.502 , 0 )
(-0.562 , 0 ) (-0.502 , 0 )
( 0.537 , 0 ) ( 0.482 , 0 )
(-0.507 , 0 ) (-0.482 , 0 )
( 0.500 , 0 ) ( 0.480 , 0 )
(-0.500 , 0 ) (-0.480 , 0 )
( 0.494 , 0 ) ( 0.476 , 0 )
( 0.463 , 0 ) (-0.476 , 0 )
( 0.453 , 0 ) ( 0.421 , 0 )
(-0.433 , 0 ) (-0.421 , 0 )
( 0.432 , 0 ) ( 0.404 , 0 )
(-0.420 , 0 ) (-0.404 , 0 )
( 0.408 , 0 ) ( 0.365 , 0 )
(-0.399 , 0 ) (-0.365 , 0 )
(-0.391 , 0 ) ( 0.362 , 0 )
( 0.364 , 0 ) (-0.362 , 0 )
( 0.361 , 0 ) (-0.333 , 0 )
(-0.359 , 0 ) ( 0.333 , 0 )
( 0.346 , 0 ) ( 0.327 , 0 )
( 0.337 , 0 ) (-0.327 , 0 )
(-0.326 , 0 ) (-0.325 , 0 )
( 0.317 , 0 ) ( 0.325 , 0 )
(-0.287 , 0 ) (-0.268 , 0 )
( 0.287 , 0 ) ( 0.268 , 0 )
( 0.280 , 0 ) (-0.266 , 0 )
(-0.272 , 0 ) ( 0.266 , 0 )
(-0.269 , 0 ) ( 0.226 , 0 )
( 0.244 , 0 ) (-0.226 , 0 )
(-0.228 , 0 ) ( 0.223 , 0 )
(-0.225 , 0 ) (-0.223 , 0 )
(-0.220 , 0 ) (-0.208 , 0 )
( 0.213 , 0 ) ( 0.208 , 0 )
( 0.200 , 0 ) ( 0.197 , 0 )
( 0.200 , 0 ) (-0.197 , 0 )
( 0.198 , 0 ) ( 0.186 , 0 )
( 0.188 , 0 ) (-0.186 , 0 )
( 0.163 , 0 ) ( 0.173 , 0 )
(-0.163 , 0 ) (-0.173 , 0 )
( 0.158 , 0 ) ( 0.148 , 0 )
(-0.139 , 0 ) (-0.148 , 0 )
(-0.133 , 0 ) ( 0.127 , 0 )
(-0.124 , 0 ) (-0.127 , 0 )
( 0.117 , 0 ) (-0.862E-01, 0 )
( 0.111 , 0 ) ( 0.862E-01, 0 )
(-0.103 , 0 ) (-0.834E-01, 0 )
(-0.958E-01, 0 ) ( 0.834E-01, 0 )
(-0.948E-01, 0 ) (-0.771E-01, 0 )
( 0.869E-01, 0 ) ( 0.771E-01, 0 )
( 0.713E-01, 0 ) (-0.752E-01, 0 )
( 0.675E-01, 0 ) ( 0.752E-01, 0 )
( 0.673E-01, 0 ) (-0.712E-01, 0 )
(-0.617E-01, 0 ) ( 0.712E-01, 0 )
( 0.567E-01, 0 ) ( 0.549E-01, 0 )
( 0.495E-01, 0 ) (-0.549E-01, 0 )
(-0.301E-01, 0 ) (-0.532E-01, 0 )
( 0.246E-01, 0 ) ( 0.532E-01, 0 )
( 0.244E-01, 0 ) ( 0.335E-01, 0 )
( 0.222E-01, 0 ) (-0.335E-01, 0 )
(-0.194E-01, 0 ) ( 0.161E-01, 0 )
(-0.165E-01, 0 ) (-0.161E-01, 0 )
(-0.794E-02, 0 ) ( 0.422E-02, 0 )
( 0.841E-09, 0 ) (-0.422E-02, 0 )
(-0.841E-09, 0 )
(-0.308E-13, 0 )
( 0.297E-14, 0.401E-15)
( 0.297E-14,-0.401E-15)
(-0.245E-14, 0 )
( 0.124E-14, 0 )
(-0.702E-15, 0.693E-15)
(-0.702E-15,-0.693E-15)
( 0.671E-15, 0.438E-15)
( 0.671E-15,-0.438E-15)
(-0.404E-15, 0.517E-15)
(-0.404E-15,-0.517E-15)
( 0.230E-15, 0.290E-15)
( 0.230E-15,-0.290E-15)
(-0.317E-15, 0 )
(-0.302E-15, 0 )
(-0.589E-16, 0.240E-15)
(-0.589E-16,-0.240E-15)
(-0.138E-15, 0.790E-16)
(-0.138E-15,-0.790E-16)
( 0.146E-15, 0 )
(-0.599E-16, 0.168E-16)
(-0.599E-16,-0.168E-16)
( 0.309E-16, 0 )
( 0.123E-16, 0 )
(-0.442E-17, 0.108E-16)
(-0.442E-17,-0.108E-16)
(-0.940E-30, 0 )
( 0 , 0 )
( 0 , 0 )
( 0 , 0 )
Resumo
- Condição de Dirichlet em todos os lados
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02} λ(91..132)=1.0
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02}
- Condição de Neumann em todos os lados
λ( 1.. 90)= { 1, 0.976, 0.970, +-0.945,..., +-0.568E-02 } λ(91..132)= (eps,...,eps**2,...,0)
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02}
- Condição de Neumann em todos os lados e subtraindo P(1,2)
λ={0.976, 0.97,+-0.945,0.905,0.883,...,+-0.568E-2,-0.44E-14} U {eps,...0}
λ={-0.955,0.911,+-0.896,+-0.884,...,0.523E-02,-0.422E-02, 0.0)
- Condição de Neumann em todos os lados fixando um ponto, P(1,2)=1
λ( 1.. 90)={+- 0.955,..., +- 0.422E-02}
λ(1)=1 λ(2..91)=( 0.999, 0.974, 0.967,-0.945, 0.939,0.902...-0.615E-02, 0.500E-02 ) λ=(eps,...0)
- Condição de Neumann em 3 lados fixando P=1 na entrada (ou na saída)
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830
lambda(n)= 0.000000000000000 0 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
λ(1..11)=1
( 0.994 , 0 ) (-0.955 , 0 )
( 0.964 , 0 ) ( 0.955 , 0 )
( 0.950 , 0 ) (-0.896 , 0 )
(-0.948 , 0 ) ( 0.896 , 0 )
( 0.920 , 0 ) ( 0.884 , 0 )
(-0.883 , 0 ) (-0.884 , 0 )
. . .
(-0.194E-01, 0 ) ( 0.161E-01, 0 )
(-0.165E-01, 0 ) (-0.161E-01, 0 )
(-0.794E-02, 0 ) ( 0.422E-02, 0 )
( 0.841E-09, 0 ) (-0.422E-02, 0 )
(-0.841E-09, 0 )
(-0.308E-13, 0 )
( 0.297E-14, 0.401E-15)
. . .
( 0 , 0 )
