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>
Condição de Neumann em todos os lados
O espectro é real com espectro σ⊂(-1,1).
Incluindo CC na matriz Eliminando CC na matriz lambda(1) = 1.0000E+00 lambda(1)= 0.955274744954830 lambda(n) = 0.0000E+00 lambda(n)= 4.218228659674831E-003 ( 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 fixando um ponto, P(1,2)=1
O espectro tem autovalores próximos do eixo real (talvez devido a erros de ponto flutuante) com σ⊂ 1.6537 ∪ (-1,1) incluindo 0.
Condição de Neumann em todos os lados e subtraindo P(1,2)
Usando Pnew=Pnew-P(1,2) temos o espectro
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, 0.000E+00) (-0.955E+00, 0.000E+00) ( 0.970E+00, 0.000E+00) ( 0.911E+00, 0.000E+00) ( 0.945E+00, 0.000E+00) ( 0.896E+00, 0.000E+00) (-0.945E+00, 0.000E+00) (-0.896E+00, 0.000E+00) ( 0.905E+00, 0.000E+00) ( 0.884E+00, 0.000E+00) ( 0.883E+00, 0.000E+00) (-0.884E+00, 0.000E+00) (-0.874E+00, 0.000E+00) (-0.825E+00, 0.000E+00) ( 0.874E+00, 0.000E+00) ( 0.825E+00, 0.000E+00) ( 0.859E+00, 0.000E+00) (-0.803E+00, 0.000E+00) (-0.859E+00, 0.000E+00) ( 0.789E+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) (-0.712E+00, 0.000E+00) ( 0.750E+00, 0.000E+00) ( 0.693E+00, 0.000E+00) (-0.726E+00, 0.000E+00) ( 0.683E+00, 0.000E+00) ( 0.726E+00, 0.000E+00) (-0.682E+00, 0.000E+00) (-0.677E+00, 0.000E+00) (-0.634E+00, 0.000E+00) ( 0.677E+00, 0.000E+00) ( 0.634E+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) ( 0.583E+00, 0.000E+00) ( 0.624E+00, 0.000E+00) (-0.575E+00, 0.000E+00) ( 0.587E+00, 0.000E+00) ( 0.575E+00, 0.000E+00) ( 0.562E+00, 0.000E+00) (-0.547E+00, 0.000E+00) (-0.562E+00, 0.000E+00) ( 0.507E+00, 0.000E+00) ( 0.544E+00, 0.000E+00) ( 0.502E+00, 0.000E+00) (-0.544E+00, 0.000E+00) (-0.494E+00, 0.000E+00) (-0.538E+00, 0.000E+00) ( 0.482E+00, 0.000E+00) ( 0.538E+00, 0.000E+00) (-0.482E+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) (-0.416E+00, 0.000E+00) ( 0.413E+00, 0.000E+00) ( 0.412E+00, 0.000E+00) (-0.405E+00, 0.000E+00) ( 0.404E+00, 0.000E+00) ( 0.405E+00, 0.000E+00) (-0.402E+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) ( 0.343E+00, 0.000E+00) (-0.381E+00, 0.000E+00) ( 0.333E+00, 0.000E+00) ( 0.381E+00, 0.000E+00) (-0.333E+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) ( 0.283E+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) (-0.218E+00, 0.000E+00) ( 0.229E+00, 0.000E+00) ( 0.208E+00, 0.000E+00) (-0.226E+00, 0.000E+00) ( 0.202E+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) (-0.190E+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) (-0.146E+00, 0.000E+00) ( 0.117E+00, 0.000E+00) ( 0.127E+00, 0.000E+00) (-0.955E-01, 0.000E+00) ( 0.122E+00, 0.000E+00) ( 0.955E-01, 0.411E-15) (-0.120E+00, 0.000E+00) ( 0.955E-01,-0.411E-15) (-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) ( 0.789E-01, 0.000E+00) ( 0.891E-01, 0.000E+00) (-0.771E-01, 0.000E+00) ( 0.868E-01, 0.000E+00) ( 0.771E-01, 0.000E+00) (-0.868E-01, 0.000E+00) (-0.752E-01, 0.000E+00) (-0.677E-01, 0.000E+00) ( 0.752E-01, 0.000E+00) ( 0.677E-01, 0.000E+00) (-0.712E-01, 0.000E+00) ( 0.653E-01, 0.000E+00) ( 0.573E-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) ( 0.395E-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) ( 0.523E-02, 0.000E+00) (-0.568E-02, 0.000E+00) (-0.422E-02, 0.000E+00) (-0.444E-14, 0.000E+00) ( 0.000E+00, 0.000E+00) ( 0.214E-14, 0.128E-14) ( 0.214E-14,-0.128E-14) ( 0.217E-14, 0.000E+00) (-0.168E-14, 0.000E+00) (-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, 0.000E+00) (-0.773E-15, 0.208E-15) (-0.773E-15,-0.208E-15) ( 0.587E-15, 0.000E+00) (-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, 0.000E+00) (-0.955E-28, 0.000E+00) (-0.281E-30, 0.325E-30) (-0.281E-30,-0.325E-30) ( 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, 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 ⊂ 11.23 ∪ (-1,1) incluindo 0.
Teste 4
- O espectro do problema não depende de dt, Re ou U0. Depende apenas do coeficientes de P (que dependem de dx,dy
- Aplicando as condições de contorno e também subtraindo uma constante de toda a solução, por exemplo, P=P-P(1,2)
