Análise espectral e estabilidade
De WikiLICC
A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo.
Vamos relatar um estudo para um problema específico.
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-2a \,\!</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.
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
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
(-0.955E+00, 0.000E+00) (-0.955E+00, 0.000E+00)
( 0.955E+00, 0.000E+00) ( 0.955E+00, 0.000E+00)
( 0.896E+00, 0.000E+00) (-0.896E+00, 0.000E+00)
(-0.896E+00, 0.000E+00) ( 0.896E+00, 0.000E+00)
(-0.884E+00, 0.000E+00) ( 0.884E+00, 0.000E+00)
( 0.884E+00, 0.000E+00) (-0.884E+00, 0.000E+00)
( 0.825E+00, 0.000E+00) ( 0.825E+00, 0.000E+00)
(-0.825E+00, 0.000E+00) (-0.825E+00, 0.000E+00)
( 0.803E+00, 0.000E+00) (-0.803E+00, 0.000E+00)
(-0.803E+00, 0.000E+00) ( 0.803E+00, 0.000E+00)
(-0.774E+00, 0.000E+00) ( 0.774E+00, 0.000E+00)
( 0.774E+00, 0.000E+00) (-0.774E+00, 0.000E+00)
( 0.732E+00, 0.000E+00) ( 0.732E+00, 0.000E+00)
(-0.732E+00, 0.000E+00) (-0.732E+00, 0.000E+00)
(-0.715E+00, 0.000E+00) ( 0.715E+00, 0.000E+00)
( 0.715E+00, 0.000E+00) (-0.715E+00, 0.000E+00)
( 0.683E+00, 0.000E+00) ( 0.683E+00, 0.000E+00)
(-0.683E+00, 0.000E+00) (-0.683E+00, 0.000E+00)
(-0.634E+00, 0.000E+00) (-0.634E+00, 0.000E+00)
( 0.634E+00, 0.000E+00) ( 0.634E+00, 0.000E+00)
( 0.621E+00, 0.000E+00) ( 0.621E+00, 0.000E+00)
(-0.621E+00, 0.000E+00) (-0.621E+00, 0.000E+00)
(-0.612E+00, 0.000E+00) ( 0.612E+00, 0.000E+00)
( 0.612E+00, 0.000E+00) (-0.612E+00, 0.000E+00)
( 0.575E+00, 0.000E+00) ( 0.575E+00, 0.000E+00)
(-0.575E+00, 0.000E+00) (-0.575E+00, 0.000E+00)
(-0.547E+00, 0.000E+00) ( 0.547E+00, 0.000E+00)
( 0.547E+00, 0.000E+00) (-0.547E+00, 0.000E+00)
( 0.502E+00, 0.000E+00) ( 0.502E+00, 0.000E+00)
(-0.502E+00, 0.000E+00) (-0.502E+00, 0.000E+00)
(-0.482E+00, 0.000E+00) ( 0.482E+00, 0.000E+00)
( 0.482E+00, 0.000E+00) (-0.482E+00, 0.000E+00)
(-0.480E+00, 0.000E+00) ( 0.480E+00, 0.000E+00)
( 0.480E+00, 0.000E+00) (-0.480E+00, 0.000E+00)
(-0.476E+00, 0.000E+00) ( 0.476E+00, 0.000E+00)
( 0.476E+00, 0.000E+00) (-0.476E+00, 0.000E+00)
( 0.421E+00, 0.000E+00) ( 0.421E+00, 0.000E+00)
(-0.421E+00, 0.000E+00) (-0.421E+00, 0.000E+00)
(-0.404E+00, 0.000E+00) ( 0.404E+00, 0.000E+00)
( 0.404E+00, 0.000E+00) (-0.404E+00, 0.000E+00)
(-0.365E+00, 0.000E+00) ( 0.365E+00, 0.000E+00)
( 0.365E+00, 0.000E+00) (-0.365E+00, 0.000E+00)
( 0.362E+00, 0.000E+00) ( 0.362E+00, 0.000E+00)
(-0.362E+00, 0.000E+00) (-0.362E+00, 0.000E+00)
( 0.333E+00, 0.000E+00) (-0.333E+00, 0.000E+00)
(-0.333E+00, 0.000E+00) ( 0.333E+00, 0.000E+00)
(-0.327E+00, 0.000E+00) ( 0.327E+00, 0.000E+00)
( 0.327E+00, 0.000E+00) (-0.327E+00, 0.000E+00)
( 0.325E+00, 0.000E+00) (-0.325E+00, 0.000E+00)
(-0.325E+00, 0.000E+00) ( 0.325E+00, 0.000E+00)
( 0.268E+00, 0.000E+00) (-0.268E+00, 0.000E+00)
(-0.268E+00, 0.000E+00) ( 0.268E+00, 0.000E+00)
( 0.266E+00, 0.000E+00) (-0.266E+00, 0.000E+00)
(-0.266E+00, 0.000E+00) ( 0.266E+00, 0.000E+00)
( 0.226E+00, 0.000E+00) ( 0.226E+00, 0.000E+00)
(-0.226E+00, 0.000E+00) (-0.226E+00, 0.000E+00)
(-0.223E+00, 0.000E+00) ( 0.223E+00, 0.000E+00)
( 0.223E+00, 0.000E+00) (-0.223E+00, 0.000E+00)
(-0.208E+00, 0.000E+00) (-0.208E+00, 0.000E+00)
( 0.208E+00, 0.000E+00) ( 0.208E+00, 0.000E+00)
(-0.197E+00, 0.000E+00) ( 0.197E+00, 0.000E+00)
( 0.197E+00, 0.000E+00) (-0.197E+00, 0.000E+00)
(-0.186E+00, 0.000E+00) ( 0.186E+00, 0.000E+00)
( 0.186E+00, 0.000E+00) (-0.186E+00, 0.000E+00)
( 0.173E+00, 0.000E+00) ( 0.173E+00, 0.000E+00)
(-0.173E+00, 0.000E+00) (-0.173E+00, 0.000E+00)
( 0.148E+00, 0.000E+00) ( 0.148E+00, 0.000E+00)
(-0.148E+00, 0.000E+00) (-0.148E+00, 0.000E+00)
(-0.127E+00, 0.000E+00) ( 0.127E+00, 0.000E+00)
( 0.127E+00, 0.000E+00) (-0.127E+00, 0.000E+00)
(-0.862E-01, 0.000E+00) (-0.862E-01, 0.000E+00)
( 0.862E-01, 0.000E+00) ( 0.862E-01, 0.000E+00)
(-0.834E-01, 0.000E+00) (-0.834E-01, 0.000E+00)
( 0.834E-01, 0.000E+00) ( 0.834E-01, 0.000E+00)
( 0.771E-01, 0.000E+00) (-0.771E-01, 0.000E+00)
(-0.771E-01, 0.000E+00) ( 0.771E-01, 0.000E+00)
( 0.752E-01, 0.000E+00) (-0.752E-01, 0.000E+00)
(-0.752E-01, 0.000E+00) ( 0.752E-01, 0.000E+00)
(-0.712E-01, 0.000E+00) (-0.712E-01, 0.000E+00)
( 0.712E-01, 0.000E+00) ( 0.712E-01, 0.000E+00)
(-0.549E-01, 0.000E+00) ( 0.549E-01, 0.000E+00)
( 0.549E-01, 0.000E+00) (-0.549E-01, 0.000E+00)
( 0.532E-01, 0.000E+00) (-0.532E-01, 0.000E+00)
(-0.532E-01, 0.000E+00) ( 0.532E-01, 0.000E+00)
(-0.335E-01, 0.000E+00) ( 0.335E-01, 0.000E+00)
( 0.335E-01, 0.000E+00) (-0.335E-01, 0.000E+00)
(-0.161E-01, 0.000E+00) ( 0.161E-01, 0.000E+00)
( 0.161E-01, 0.000E+00) (-0.161E-01, 0.000E+00)
( 0.422E-02, 0.000E+00) ( 0.422E-02, 0.000E+00)
(-0.422E-02, 0.000E+00) (-0.422E-02, 0.000E+00)
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)= 0.000000000000000E+000 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 0.100E+01, 0.000E+00) (-0.955E+00, 0.000E+00)
( 0.976E+00, 0.000E+00) ( 0.955E+00, 0.000E+00)
( 0.970E+00, 0.000E+00) (-0.896E+00, 0.000E+00)
(-0.945E+00, 0.000E+00) ( 0.896E+00, 0.000E+00)
( 0.945E+00, 0.000E+00) ( 0.884E+00, 0.000E+00)
( 0.905E+00, 0.000E+00) (-0.884E+00, 0.000E+00)
( 0.883E+00, 0.000E+00) ( 0.825E+00, 0.000E+00)
(-0.874E+00, 0.000E+00) (-0.825E+00, 0.000E+00)
( 0.874E+00, 0.000E+00) (-0.803E+00, 0.000E+00)
( 0.859E+00, 0.000E+00) ( 0.803E+00, 0.000E+00)
(-0.859E+00, 0.000E+00) ( 0.774E+00, 0.000E+00)
( 0.794E+00, 0.000E+00) (-0.774E+00, 0.000E+00)
(-0.788E+00, 0.000E+00) ( 0.732E+00, 0.000E+00)
( 0.788E+00, 0.000E+00) (-0.732E+00, 0.000E+00)
( 0.764E+00, 0.000E+00) ( 0.715E+00, 0.000E+00)
(-0.764E+00, 0.000E+00) (-0.715E+00, 0.000E+00)
( 0.750E+00, 0.000E+00) ( 0.683E+00, 0.000E+00)
(-0.726E+00, 0.000E+00) (-0.683E+00, 0.000E+00)
( 0.726E+00, 0.000E+00) (-0.634E+00, 0.000E+00)
( 0.677E+00, 0.000E+00) ( 0.634E+00, 0.000E+00)
(-0.677E+00, 0.000E+00) ( 0.621E+00, 0.000E+00)
(-0.655E+00, 0.000E+00) (-0.621E+00, 0.000E+00)
( 0.655E+00, 0.000E+00) ( 0.612E+00, 0.000E+00)
( 0.655E+00, 0.000E+00) (-0.612E+00, 0.000E+00)
( 0.624E+00, 0.000E+00) ( 0.575E+00, 0.000E+00)
(-0.624E+00, 0.000E+00) (-0.575E+00, 0.000E+00)
( 0.587E+00, 0.000E+00) ( 0.547E+00, 0.000E+00)
(-0.562E+00, 0.000E+00) (-0.547E+00, 0.000E+00)
( 0.562E+00, 0.000E+00) ( 0.502E+00, 0.000E+00)
(-0.544E+00, 0.000E+00) (-0.502E+00, 0.000E+00)
( 0.544E+00, 0.000E+00) ( 0.482E+00, 0.000E+00)
( 0.538E+00, 0.000E+00) (-0.482E+00, 0.000E+00)
(-0.538E+00, 0.000E+00) ( 0.480E+00, 0.000E+00)
( 0.500E+00, 0.000E+00) (-0.480E+00, 0.000E+00)
( 0.491E+00, 0.000E+00) ( 0.476E+00, 0.000E+00)
(-0.491E+00, 0.000E+00) (-0.476E+00, 0.000E+00)
(-0.470E+00, 0.000E+00) ( 0.421E+00, 0.000E+00)
( 0.470E+00, 0.000E+00) (-0.421E+00, 0.000E+00)
( 0.413E+00, 0.000E+00) ( 0.404E+00, 0.000E+00)
(-0.405E+00, 0.000E+00) (-0.404E+00, 0.000E+00)
( 0.405E+00, 0.000E+00) ( 0.365E+00, 0.000E+00)
(-0.389E+00, 0.000E+00) (-0.365E+00, 0.000E+00)
( 0.389E+00, 0.000E+00) ( 0.362E+00, 0.000E+00)
(-0.383E+00, 0.000E+00) (-0.362E+00, 0.000E+00)
( 0.383E+00, 0.000E+00) (-0.333E+00, 0.000E+00)
(-0.381E+00, 0.000E+00) ( 0.333E+00, 0.000E+00)
( 0.381E+00, 0.000E+00) ( 0.327E+00, 0.000E+00)
( 0.345E+00, 0.000E+00) (-0.327E+00, 0.000E+00)
(-0.318E+00, 0.000E+00) (-0.325E+00, 0.000E+00)
( 0.318E+00, 0.000E+00) ( 0.325E+00, 0.000E+00)
( 0.315E+00, 0.000E+00) (-0.268E+00, 0.000E+00)
(-0.315E+00, 0.000E+00) ( 0.268E+00, 0.000E+00)
(-0.250E+00, 0.000E+00) (-0.266E+00, 0.000E+00)
( 0.250E+00, 0.000E+00) ( 0.266E+00, 0.000E+00)
( 0.250E+00, 0.000E+00) ( 0.226E+00, 0.000E+00)
(-0.241E+00, 0.000E+00) (-0.226E+00, 0.000E+00)
( 0.241E+00, 0.000E+00) ( 0.223E+00, 0.000E+00)
(-0.229E+00, 0.000E+00) (-0.223E+00, 0.000E+00)
( 0.229E+00, 0.000E+00) (-0.208E+00, 0.000E+00)
(-0.226E+00, 0.000E+00) ( 0.208E+00, 0.000E+00)
( 0.226E+00, 0.000E+00) ( 0.197E+00, 0.000E+00)
( 0.207E+00, 0.000E+00) (-0.197E+00, 0.000E+00)
(-0.207E+00, 0.000E+00) ( 0.186E+00, 0.000E+00)
( 0.206E+00, 0.000E+00) (-0.186E+00, 0.000E+00)
( 0.176E+00, 0.000E+00) ( 0.173E+00, 0.000E+00)
(-0.176E+00, 0.000E+00) (-0.173E+00, 0.000E+00)
(-0.155E+00, 0.000E+00) ( 0.148E+00, 0.000E+00)
( 0.155E+00, 0.000E+00) (-0.148E+00, 0.000E+00)
( 0.117E+00, 0.000E+00) ( 0.127E+00, 0.000E+00)
( 0.955E-01, 0.000E+00) (-0.127E+00, 0.000E+00)
( 0.955E-01, 0.000E+00) (-0.862E-01, 0.000E+00)
(-0.955E-01, 0.000E+00) ( 0.862E-01, 0.000E+00)
(-0.925E-01, 0.000E+00) (-0.834E-01, 0.000E+00)
( 0.925E-01, 0.000E+00) ( 0.834E-01, 0.000E+00)
(-0.891E-01, 0.000E+00) (-0.771E-01, 0.000E+00)
( 0.891E-01, 0.000E+00) ( 0.771E-01, 0.000E+00)
(-0.868E-01, 0.000E+00) (-0.752E-01, 0.000E+00)
( 0.868E-01, 0.000E+00) ( 0.752E-01, 0.000E+00)
( 0.677E-01, 0.000E+00) (-0.712E-01, 0.000E+00)
(-0.677E-01, 0.000E+00) ( 0.712E-01, 0.000E+00)
(-0.653E-01, 0.000E+00) ( 0.549E-01, 0.000E+00)
( 0.653E-01, 0.000E+00) (-0.549E-01, 0.000E+00)
( 0.439E-01, 0.000E+00) (-0.532E-01, 0.000E+00)
(-0.439E-01, 0.000E+00) ( 0.532E-01, 0.000E+00)
( 0.302E-01, 0.000E+00) ( 0.335E-01, 0.000E+00)
( 0.245E-01, 0.000E+00) (-0.335E-01, 0.000E+00)
( 0.215E-01, 0.000E+00) ( 0.161E-01, 0.000E+00)
(-0.215E-01, 0.000E+00) (-0.161E-01, 0.000E+00)
(-0.568E-02, 0.000E+00) ( 0.422E-02, 0.000E+00)
( 0.568E-02, 0.000E+00) (-0.422E-02, 0.000E+00)
( 0.251E-14, 0.857E-15)
( 0.251E-14,-0.857E-15)
(-0.255E-14, 0.000E+00)
( 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, 0.000E+00)
(-0.772E-15, 0.000E+00)
( 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, 0.000E+00)
(-0.433E-15, 0.000E+00)
(-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, 0.000E+00)
(-0.230E-16, 0.000E+00)
(-0.824E-30, 0.885E-30)
(-0.824E-30,-0.885E-30)
( 0.680E-30, 0.000E+00)
(-0.684E-31, 0.000E+00)
( 0.317E-31, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
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)= 0.000000000000000E+000 lambda(n)= 0.000000000000000E+000 ( 0.976E+00, 0.0 ) (-0.955E+00, 0.0 ) ( 0.970E+00, 0.0 ) ( 0.911E+00, 0.0 ) ( 0.945E+00, 0.0 ) ( 0.896E+00, 0.0 ) (-0.945E+00, 0.0 ) (-0.896E+00, 0.0 ) ( 0.905E+00, 0.0 ) ( 0.884E+00, 0.0 ) ( 0.883E+00, 0.0 ) (-0.884E+00, 0.0 ) (-0.874E+00, 0.0 ) (-0.825E+00, 0.0 ) ( 0.874E+00, 0.0 ) ( 0.825E+00, 0.0 ) ( 0.859E+00, 0.0 ) (-0.803E+00, 0.0 ) (-0.859E+00, 0.0 ) ( 0.789E+00, 0.0 ) ( 0.794E+00, 0.0 ) (-0.774E+00, 0.0 ) ( 0.788E+00, 0.0 ) (-0.732E+00, 0.0 ) (-0.788E+00, 0.0 ) ( 0.732E+00, 0.0 ) ( 0.764E+00, 0.0 ) ( 0.715E+00, 0.0 ) (-0.764E+00, 0.0 ) (-0.712E+00, 0.0 ) ( 0.750E+00, 0.0 ) ( 0.693E+00, 0.0 ) (-0.726E+00, 0.0 ) ( 0.683E+00, 0.0 ) ( 0.726E+00, 0.0 ) (-0.682E+00, 0.0 ) (-0.677E+00, 0.0 ) (-0.634E+00, 0.0 ) ( 0.677E+00, 0.0 ) ( 0.634E+00, 0.0 ) ( 0.655E+00, 0.0 ) (-0.621E+00, 0.0 ) (-0.655E+00, 0.0 ) (-0.612E+00, 0.0 ) ( 0.655E+00, 0.0 ) ( 0.612E+00, 0.0 ) (-0.624E+00, 0.0 ) ( 0.583E+00, 0.0 ) ( 0.624E+00, 0.0 ) (-0.575E+00, 0.0 ) ( 0.587E+00, 0.0 ) ( 0.575E+00, 0.0 ) ( 0.562E+00, 0.0 ) (-0.547E+00, 0.0 ) (-0.562E+00, 0.0 ) ( 0.507E+00, 0.0 ) ( 0.544E+00, 0.0 ) ( 0.502E+00, 0.0 ) (-0.544E+00, 0.0 ) (-0.494E+00, 0.0 ) (-0.538E+00, 0.0 ) ( 0.482E+00, 0.0 ) ( 0.538E+00, 0.0 ) (-0.482E+00, 0.0 ) ( 0.500E+00, 0.0 ) (-0.480E+00, 0.0 ) ( 0.491E+00, 0.0 ) (-0.476E+00, 0.0 ) (-0.491E+00, 0.0 ) ( 0.476E+00, 0.0 ) (-0.470E+00, 0.0 ) ( 0.421E+00, 0.0 ) ( 0.470E+00, 0.0 ) (-0.416E+00, 0.0 ) ( 0.413E+00, 0.0 ) ( 0.412E+00, 0.0 ) (-0.405E+00, 0.0 ) ( 0.404E+00, 0.0 ) ( 0.405E+00, 0.0 ) (-0.402E+00, 0.0 ) (-0.389E+00, 0.0 ) (-0.365E+00, 0.0 ) ( 0.389E+00, 0.0 ) (-0.362E+00, 0.0 ) ( 0.383E+00, 0.0 ) ( 0.362E+00, 0.0 ) (-0.383E+00, 0.0 ) ( 0.343E+00, 0.0 ) (-0.381E+00, 0.0 ) ( 0.333E+00, 0.0 ) ( 0.381E+00, 0.0 ) (-0.333E+00, 0.0 ) ( 0.345E+00, 0.0 ) (-0.327E+00, 0.0 ) ( 0.318E+00, 0.0 ) (-0.325E+00, 0.0 ) (-0.318E+00, 0.0 ) ( 0.325E+00, 0.0 ) ( 0.315E+00, 0.0 ) ( 0.283E+00, 0.0 ) (-0.315E+00, 0.0 ) (-0.268E+00, 0.0 ) ( 0.250E+00, 0.0 ) ( 0.266E+00, 0.0 ) (-0.250E+00, 0.0 ) (-0.266E+00, 0.0 ) ( 0.250E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.241E+00, 0.0 ) ( 0.226E+00, 0.0 ) (-0.241E+00, 0.0 ) ( 0.223E+00, 0.0 ) (-0.229E+00, 0.0 ) (-0.218E+00, 0.0 ) ( 0.229E+00, 0.0 ) ( 0.208E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.202E+00, 0.0 ) ( 0.226E+00, 0.0 ) (-0.197E+00, 0.0 ) ( 0.207E+00, 0.0 ) ( 0.197E+00, 0.0 ) (-0.207E+00, 0.0 ) (-0.190E+00, 0.0 ) ( 0.206E+00, 0.0 ) (-0.186E+00, 0.0 ) ( 0.176E+00, 0.0 ) (-0.173E+00, 0.0 ) (-0.176E+00, 0.0 ) ( 0.173E+00, 0.0 ) ( 0.155E+00, 0.0 ) ( 0.148E+00, 0.0 ) (-0.155E+00, 0.0 ) (-0.146E+00, 0.0 ) ( 0.117E+00, 0.0 ) ( 0.127E+00, 0.0 ) (-0.955E-01, 0.0 ) ( 0.122E+00, 0.0 ) ( 0.955E-01, 0.411E-15) (-0.120E+00, 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)= 0.000000000000000E+000 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 0.100E+01, 0.000E+00) (-0.955E+00, 0.000E+00)
( 0.999E+00, 0.000E+00) ( 0.955E+00, 0.000E+00)
( 0.974E+00, 0.000E+00) (-0.896E+00, 0.000E+00)
( 0.967E+00, 0.000E+00) ( 0.896E+00, 0.000E+00)
(-0.945E+00, 0.000E+00) ( 0.884E+00, 0.000E+00)
( 0.939E+00, 0.000E+00) (-0.884E+00, 0.000E+00)
( 0.902E+00, 0.000E+00) ( 0.825E+00, 0.000E+00)
( 0.881E+00, 0.000E+00) (-0.825E+00, 0.000E+00)
(-0.874E+00, 0.000E+00) (-0.803E+00, 0.000E+00)
( 0.869E+00, 0.000E+00) ( 0.803E+00, 0.000E+00)
(-0.859E+00, 0.000E+00) ( 0.774E+00, 0.000E+00)
( 0.849E+00, 0.000E+00) (-0.774E+00, 0.000E+00)
( 0.793E+00, 0.000E+00) ( 0.732E+00, 0.000E+00)
(-0.788E+00, 0.000E+00) (-0.732E+00, 0.000E+00)
( 0.782E+00, 0.000E+00) ( 0.715E+00, 0.000E+00)
(-0.764E+00, 0.000E+00) (-0.715E+00, 0.000E+00)
( 0.758E+00, 0.000E+00) ( 0.683E+00, 0.000E+00)
( 0.746E+00, 0.000E+00) (-0.683E+00, 0.000E+00)
(-0.726E+00, 0.000E+00) (-0.634E+00, 0.000E+00)
( 0.718E+00, 0.000E+00) ( 0.634E+00, 0.000E+00)
(-0.677E+00, 0.000E+00) ( 0.621E+00, 0.000E+00)
( 0.672E+00, 0.000E+00) (-0.621E+00, 0.000E+00)
(-0.655E+00, 0.000E+00) ( 0.612E+00, 0.000E+00)
( 0.655E+00, 0.000E+00) (-0.612E+00, 0.000E+00)
( 0.645E+00, 0.000E+00) ( 0.575E+00, 0.000E+00)
(-0.625E+00, 0.000E+00) (-0.575E+00, 0.000E+00)
( 0.617E+00, 0.000E+00) ( 0.547E+00, 0.000E+00)
( 0.584E+00, 0.000E+00) (-0.547E+00, 0.000E+00)
(-0.563E+00, 0.000E+00) ( 0.502E+00, 0.000E+00)
( 0.558E+00, 0.000E+00) (-0.502E+00, 0.000E+00)
(-0.545E+00, 0.000E+00) ( 0.482E+00, 0.000E+00)
( 0.541E+00, 0.000E+00) (-0.482E+00, 0.000E+00)
(-0.538E+00, 0.000E+00) ( 0.480E+00, 0.000E+00)
( 0.527E+00, 0.000E+00) (-0.480E+00, 0.000E+00)
( 0.498E+00, 0.000E+00) ( 0.476E+00, 0.000E+00)
(-0.492E+00, 0.000E+00) (-0.476E+00, 0.000E+00)
( 0.485E+00, 0.000E+00) ( 0.421E+00, 0.000E+00)
(-0.470E+00, 0.000E+00) (-0.421E+00, 0.000E+00)
( 0.462E+00, 0.000E+00) ( 0.404E+00, 0.000E+00)
( 0.412E+00, 0.000E+00) (-0.404E+00, 0.000E+00)
(-0.406E+00, 0.000E+00) ( 0.365E+00, 0.000E+00)
( 0.401E+00, 0.000E+00) (-0.365E+00, 0.000E+00)
(-0.389E+00, 0.000E+00) ( 0.362E+00, 0.000E+00)
( 0.387E+00, 0.000E+00) (-0.362E+00, 0.000E+00)
(-0.384E+00, 0.000E+00) (-0.333E+00, 0.000E+00)
( 0.382E+00, 0.000E+00) ( 0.333E+00, 0.000E+00)
(-0.382E+00, 0.000E+00) ( 0.327E+00, 0.000E+00)
( 0.366E+00, 0.000E+00) (-0.327E+00, 0.000E+00)
( 0.343E+00, 0.000E+00) (-0.325E+00, 0.000E+00)
(-0.319E+00, 0.000E+00) ( 0.325E+00, 0.000E+00)
( 0.317E+00, 0.000E+00) (-0.268E+00, 0.000E+00)
(-0.316E+00, 0.000E+00) ( 0.268E+00, 0.000E+00)
( 0.306E+00, 0.000E+00) (-0.266E+00, 0.000E+00)
(-0.253E+00, 0.000E+00) ( 0.266E+00, 0.000E+00)
( 0.250E+00, 0.000E+00) ( 0.226E+00, 0.000E+00)
( 0.247E+00, 0.000E+00) (-0.226E+00, 0.000E+00)
(-0.243E+00, 0.000E+00) ( 0.223E+00, 0.000E+00)
( 0.237E+00, 0.000E+00) (-0.223E+00, 0.000E+00)
(-0.230E+00, 0.000E+00) (-0.208E+00, 0.000E+00)
( 0.227E+00, 0.000E+00) ( 0.208E+00, 0.000E+00)
(-0.226E+00, 0.000E+00) ( 0.197E+00, 0.000E+00)
( 0.219E+00, 0.000E+00) (-0.197E+00, 0.000E+00)
(-0.209E+00, 0.000E+00) ( 0.186E+00, 0.000E+00)
( 0.206E+00, 0.000E+00) (-0.186E+00, 0.000E+00)
( 0.197E+00, 0.000E+00) ( 0.173E+00, 0.000E+00)
(-0.176E+00, 0.000E+00) (-0.173E+00, 0.000E+00)
( 0.172E+00, 0.000E+00) ( 0.148E+00, 0.000E+00)
(-0.156E+00, 0.000E+00) (-0.148E+00, 0.000E+00)
( 0.150E+00, 0.000E+00) ( 0.127E+00, 0.000E+00)
( 0.116E+00, 0.000E+00) (-0.127E+00, 0.000E+00)
(-0.102E+00, 0.000E+00) (-0.862E-01, 0.000E+00)
( 0.955E-01, 0.000E+00) ( 0.862E-01, 0.000E+00)
( 0.941E-01, 0.000E+00) (-0.834E-01, 0.000E+00)
(-0.930E-01, 0.000E+00) ( 0.834E-01, 0.000E+00)
( 0.916E-01, 0.000E+00) (-0.771E-01, 0.000E+00)
(-0.911E-01, 0.000E+00) ( 0.771E-01, 0.000E+00)
( 0.882E-01, 0.000E+00) (-0.752E-01, 0.000E+00)
(-0.881E-01, 0.000E+00) ( 0.752E-01, 0.000E+00)
( 0.795E-01, 0.000E+00) (-0.712E-01, 0.000E+00)
(-0.708E-01, 0.000E+00) ( 0.712E-01, 0.000E+00)
( 0.660E-01, 0.000E+00) ( 0.549E-01, 0.000E+00)
(-0.655E-01, 0.000E+00) (-0.549E-01, 0.000E+00)
( 0.587E-01, 0.000E+00) (-0.532E-01, 0.000E+00)
(-0.463E-01, 0.000E+00) ( 0.532E-01, 0.000E+00)
( 0.389E-01, 0.000E+00) ( 0.335E-01, 0.000E+00)
( 0.298E-01, 0.000E+00) (-0.335E-01, 0.000E+00)
( 0.243E-01, 0.000E+00) ( 0.161E-01, 0.000E+00)
(-0.229E-01, 0.000E+00) (-0.161E-01, 0.000E+00)
( 0.186E-01, 0.000E+00) ( 0.422E-02, 0.000E+00)
(-0.615E-02, 0.000E+00) (-0.422E-02, 0.000E+00)
( 0.500E-02, 0.000E+00)
(-0.799E-14, 0.000E+00)
( 0.595E-14, 0.000E+00)
(-0.228E-14, 0.000E+00)
(-0.173E-14, 0.900E-15)
(-0.173E-14,-0.900E-15)
( 0.176E-14, 0.000E+00)
( 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, 0.000E+00)
(-0.694E-15, 0.000E+00)
(-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, 0.000E+00)
( 0.157E-15, 0.000E+00)
( 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, 0.000E+00)
( 0.172E-17, 0.943E-17)
( 0.172E-17,-0.943E-17)
( 0.454E-26, 0.000E+00)
(-0.800E-30, 0.213E-30)
(-0.800E-30,-0.213E-30)
( 0.683E-30, 0.000E+00)
(-0.556E-31, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
Condição de Neumann em 3 lados fixando P=1 na entrada (ou na saída)
O espectro tem a maioria dos autovalores próximos do eixo real com alguns autovalores complexos (um pequeno circulo de valores complexos dentro do círculo unitário) com σ ⊂ 11.26 ∪ (-1,1) incluindo 0.
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830
lambda(n)= 0.000000000000000E+000 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.994E+00, 0.000E+00) (-0.955E+00, 0.000E+00)
( 0.964E+00, 0.000E+00) ( 0.955E+00, 0.000E+00)
( 0.950E+00, 0.000E+00) (-0.896E+00, 0.000E+00)
(-0.948E+00, 0.000E+00) ( 0.896E+00, 0.000E+00)
( 0.920E+00, 0.000E+00) ( 0.884E+00, 0.000E+00)
(-0.883E+00, 0.000E+00) (-0.884E+00, 0.000E+00)
( 0.877E+00, 0.000E+00) ( 0.825E+00, 0.000E+00)
( 0.867E+00, 0.000E+00) (-0.825E+00, 0.000E+00)
(-0.861E+00, 0.000E+00) (-0.803E+00, 0.000E+00)
( 0.836E+00, 0.000E+00) ( 0.803E+00, 0.000E+00)
( 0.834E+00, 0.000E+00) ( 0.774E+00, 0.000E+00)
(-0.796E+00, 0.000E+00) (-0.774E+00, 0.000E+00)
(-0.782E+00, 0.000E+00) ( 0.732E+00, 0.000E+00)
( 0.750E+00, 0.000E+00) (-0.732E+00, 0.000E+00)
( 0.750E+00, 0.000E+00) ( 0.715E+00, 0.000E+00)
( 0.744E+00, 0.000E+00) (-0.715E+00, 0.000E+00)
(-0.728E+00, 0.000E+00) ( 0.683E+00, 0.000E+00)
( 0.720E+00, 0.000E+00) (-0.683E+00, 0.000E+00)
( 0.700E+00, 0.000E+00) (-0.634E+00, 0.000E+00)
(-0.695E+00, 0.000E+00) ( 0.634E+00, 0.000E+00)
(-0.663E+00, 0.000E+00) ( 0.621E+00, 0.000E+00)
(-0.653E+00, 0.000E+00) (-0.621E+00, 0.000E+00)
( 0.633E+00, 0.000E+00) ( 0.612E+00, 0.000E+00)
( 0.617E+00, 0.000E+00) (-0.612E+00, 0.000E+00)
( 0.611E+00, 0.000E+00) ( 0.575E+00, 0.000E+00)
( 0.581E+00, 0.000E+00) (-0.575E+00, 0.000E+00)
( 0.581E+00, 0.000E+00) ( 0.547E+00, 0.000E+00)
(-0.566E+00, 0.000E+00) (-0.547E+00, 0.000E+00)
(-0.565E+00, 0.000E+00) ( 0.502E+00, 0.000E+00)
(-0.562E+00, 0.000E+00) (-0.502E+00, 0.000E+00)
( 0.537E+00, 0.000E+00) ( 0.482E+00, 0.000E+00)
(-0.507E+00, 0.000E+00) (-0.482E+00, 0.000E+00)
( 0.500E+00, 0.000E+00) ( 0.480E+00, 0.000E+00)
(-0.500E+00, 0.000E+00) (-0.480E+00, 0.000E+00)
( 0.494E+00, 0.000E+00) ( 0.476E+00, 0.000E+00)
( 0.463E+00, 0.000E+00) (-0.476E+00, 0.000E+00)
( 0.453E+00, 0.000E+00) ( 0.421E+00, 0.000E+00)
(-0.433E+00, 0.000E+00) (-0.421E+00, 0.000E+00)
( 0.432E+00, 0.000E+00) ( 0.404E+00, 0.000E+00)
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