Does anyone know how I can solve (using finite differences) higher order hyperbolic systems (e.g. wave equations in elasticity)? The system I want to solve looks like:

u = u(x,t)

u_tt + a(x)*u_t + b(x)*u_x + c(x)*u_xx + d(x)*u_xxx + e(x)*u_xxxx + f(x)*u = g(x,t)

_t and _x is partial differentiation with respect to t and x.

The best method is Level Set Method by Stanley Osher and James Sethian

Vinod Dhiman

Why would level set methods be of any help? I have not started thinking about boundary conditions as yet ...

Is there a way to solve an equation numerically without specifying boundary conditions. Well, it could be, I really don't know then.

Vinod Dhiman

I would suggest using DRP schemes or compact schemes.They were derived mainly for aeroacoustics but should work fine with the problem you have in hand provided since the schemes are dispersion preserving over a wide range of wave numbers. I would suggest you to look up for papers relating to them using scholar.google.com.

-H

Such equations arise in structural analysis (e.g., beam, plate and shell transient response). Actually my M.Sc. thesis was about developing a FEM for shells, where the equations may be degenerated to yours if most of the non-linearities are omitted. However, I suggest you start from a simpler approach. If you use FEM or FVM, the BC may be simple for treatment if they are of usual types (Dirichlet or Neuman). For FDM, you may need to add ghost cells to take care of this, or otherwise use one-sided differences at the boundaries, but ones which are of the same order as your scheme away from the boundaries (e.g., if your scheme is 2nd order, do NOT use the usual 1st order forward / backward schemes for the BC).

Rami: Can you tell me which scheme to use? I can't find a way to fit this into the standard hyperbolic conservation form, and so no scheme really comes to mind.

Have you checked that your equation is strictly hyperbolic? and more specifically is it well-posed for your choice of boundary conidtions and coefficients (for numerical problems a well-posed problem, in the sense of Cauchy, can appear to be ill-posed because of the growth of gridscale noise).

As an observation if e(x)>0 ( =1 say) then the dominant balance in your equation is u_tt + u_xxxx = 0 which has solutions of the form exp(+/- i.t.k^2) which oscillate rapidly with increasing t (k is the wavenumber) - the inverse transform will be a Fresnel integral. If e<0 your problem will be ill-posed since the solution grows like exp(t.k^2).

The simplest way to solve this equation is to use explicit 2nd order central differences and march in time with a small timestep.

>>u = u(x,t)

>>u_tt + a(x)*u_t + b(x)*u_x + c(x)*u_xx + d(x)*u_xxx + e(x)*u_xxxx + f(x)*u = g(x,t)

Do any of the coefficients of the u_nx terms ever become negative? If so, you will have to be careful of both 'static stability' (x-related) & 'dynamic stability' (t-related)issues...

Small-enough time steps will probably be enough to settle the dynamic stability issues, but static stability issues will put a bound on maximum dx step...

diaw...