Timeline for numerical approach for system of non-linear partial-ordinary differential equations
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 5, 2016 at 10:02 | history | edited | james506 | CC BY-SA 3.0 |
alternative solution approach
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| Jul 1, 2016 at 11:38 | comment | added | Bill Greene | I don't have any reference suggestions. As a first attempt, I would just substitute $f^2/2$ and see how it works. | |
| Jul 1, 2016 at 8:45 | comment | added | james506 | $f$ represents a non-dimensional pressure. The second derivative w.r.t time arises due to an assumption of weak acoustic non-linearity. Do you have any suggestions for references for handling the burgers term in this context? | |
| Jun 30, 2016 at 20:40 | comment | added | Bill Greene | Yes, I appreciate that getting a good spatial discretization for these hyperbolic equations is tricky, and I am aware of some of the tricks, but not my area of expertise, unfortunately. By the way, what is $f$? If it were the velocity of the fluid I would not have expected that term in eqn 1 that is a second derivative wrt time. | |
| Jun 30, 2016 at 20:21 | comment | added | james506 | I think I follow you now. So yes I guess this would then allow you to have $d/dT = rhs$. The remaining issue I see though is how to handle the Burgers term, $\partial F/\partial T$? | |
| Jun 30, 2016 at 19:57 | comment | added | Bill Greene | Well, I appreciate that solving your eqn 1, a nonlinear, hyperbolic equation with shocks has its own set of discretization issues. But if discretizing $\partial f/\partial X$ with a simple first-order backward difference (i.e. upwind) scheme, is acceptable, wouldn't that lead to an ODE system? The unknowns in the system of ODE would be $f, g, q, \xi, \psi$ at each of the FD points. You would also need a sixth unknown to convert eqn 1 to first order form. Doesn't that give a set of ODE with 6N equations (N=number of FD points)? | |
| Jun 30, 2016 at 19:19 | comment | added | james506 | @BillGreene, Would you be able to clarify how you would do this discretising in space ($X$) instead of $T$? If you discretise in $X$ I can't see how you would put the equations in the form $d/dT = rhs(X)$, which as I understand is needed for an ODE solver? | |
| Jun 30, 2016 at 18:44 | comment | added | Bill Greene | No, my suggestion is to discretize in space (X) so that you end up with a standard system of ODE to solve. The ODE system in T is a standard initial value problem that should be solvable with many black box ODE solvers. | |
| Jun 30, 2016 at 18:37 | comment | added | james506 | This is essentially what method (2) is. The equations are discretised in $T$ only, using finite differences. However, my understanding is you can only use an ODE solver which is compatible with DAEs for this system as there is no $\partial / \partial X$ terms in (2-5) | |
| Jun 30, 2016 at 18:01 | comment | added | Bill Greene | I was thinking of a fairly standard method-of-lines approach where you would discretize all five equations in the spatial dimension and then pass the resulting system of ODE to a standard solver for solution in time. | |
| Jun 30, 2016 at 17:07 | history | answered | james506 | CC BY-SA 3.0 |