How Do Complex Eigenvalues Affect Numerical Solutions of PDE Systems?

[desA]

I'm currently researching a 3d tensor, where certain combinations of terms can cause the principal values (eigenvalues) to become complex. This would then seem to imply that the associated eigenvectors would also become complex.

What now, if this tensor were part of a larger equation, ultimately solved in expanded form (3d), using numerical methods?

What could we reasonably expect to occur? Would a real solution be obtained, or would the numerics crash against the complex solutions.

If now, this tensor is dotted against a vector in such a way that the inner workings of the tensor are hidden in the detail of the expanded pde form, & the new equation is solved using numerical methods?

What could we reasonably expect to occur?

In the second form, in a few simulations I have to hand, in the final computed result field, computation of the eigenvalues shows up as real, or complex & in very distinct regions - so there is a carry-through, only it is not obvious in the fully-expanded 3d form.

There is a very sound reason behind the questions I'm asking, as it applies to a few rather well-known pde equations. I'd love to know if this hurdle has been addressed in the past & if so, a few links would be gratefully appreciated.

 

[Chris Hillman]

I think it would really help if you could be more specific about what tensor you are studying, and in what context.

 

[desA]

I think it would really help if you could be more specific about what tensor you are studying, and in what context.

I'm approaching the tensor from a fairly general point-of-view, but I'll try to narrow down the concepts a little further to the dyadic product of nabla vector & a vector, in 2D, for now.

The will produce a tensor, the terms of which are all spatial gradients. If we now search for the principle (eigenv-) values, we can find certain combinations of gradients which cause the eigenvalues to cross through into complex values. This should then provide corresponding complex eigenvectors.

If an inner product is then formed between this nabla-v tensor & its original vector, a vector results - which consists of pde terms eg.

m.del(m) with m=vector

We could then set this equal to another vector & solve the resulting system of pde's eg.

m.del(m) = q with q=vector (arbitrary) (1)

I'll write it out, although not in latex, as I have no idea how to use it.

m.del(m) = q

[m1 m2].[m1,1 m2,1] = [q1]
...[m1,2 m2,2]...[q2]

[m1.m1,1 + m2.m1,2] = [q1]
[m1.m2,1 + m2.m2,2]...[q2] ...(2)

Resulting in a system of 2 nonlinear pde's

What if we were given only eqns(2) & knew nothing of tensor character & the fact that the eigevalues of the gradient tensor can become complex. We go ahead & solve the system numerically, because we know no other way.

Questions:
1. What impact, if any, would the fact that the gradient tensor had complex eigenvalues, have on the final solution?
2. Should we expect some odd behaviour during the solution?
3. Could we expect jump solutions to occur in the maths to contain the solutions to the real domain?

Thanks for your input.

desA