Solvers for odd order PDE finite difference discretisation

Question

I am used to solving elliptic PDEs of even order. I was wondering what would one do for odd order PDEs. Notably the discretisations of those results in unsymmetric matrices. I tried solving the following simple problem:

$$\frac{d^3u}{dx^3}(x) = 0,\, x\in \Omega \quad u(x) = f(x), \, x \in \Gamma_D, \, \partial^{l}_nu(x) = 0, x\in\Gamma_N$$

I would generally expect to get some quadratic spline as the solution to the above. But I got a very oscillatory solution which I believe is due to numerical instability. I used the discretisation: $$\frac{d^3u}{dx^3} = \begin{bmatrix} -\frac{1}{2} & 1 & 0 & -1 & \frac{1}{2} \end{bmatrix} *u+O(h^2)$$

Since the system matrix $L$ is unsymmetric, I use a CGNR solver to solve $L^TLx = L^Tb$ instead, but nevertheless things don't seem to work out. I have never dealt with odd-order PDEs numerically. Are there some references discussing this? Am I supposed to use a specific discretisation like an upwind scheme? I don't think this makes any sense as the factor in front of $\frac{d^3u}{dx^3}$ seems to be irrelevant considering that the right-hand side is zero. Would the above problem be stabilized by solving:

$$\left |\frac{d^3u}{dx^3}\right | = 0,$$

instead? Clearly I cannot write the above as $y=Lx$ anymore and would have to potentially extend it to $\partial_t u = \left|\frac{d^3u}{dx^3}\right|$, e.g. apply explicit Euler and send the time to infinity. I just want to make sure whether there are known methods to handle odd order PDEs numerically before I try some stupid ideas.

Answer 1

The matrix for $d/dx$ is skew-symmetric for a suitable discretization scheme, and the same is true for its odd powers.

Moreover, the operator $i d/dx$ is hermitian, and again, by using a skew-symmetric discretization, all its powers are hermitian as well.

If you then have a PDE that only contains odd derivatives of $x$, you can consider the solution function $v(x)=i u(x)$, and bring the imaginary unit to the operator side and thus obtain a hermitian problem, which can be solved , e.g., by conjugate gradient again.

That's it for now, as I'm on my mobile phone and typing is exhausting. Just leave a comment and ask, I'll write more once I'm back at my laptop.

Answer 2

As mentioned in my comments to davidhigh's answer, using the discretisation in my question results in a skew-symmetric matrix which when modified to add Dirichlet conditions becomes singular. To better understand the problem I looked at discretisations of:

$$\frac{du}{dx} = 0$$

and those require an upwind scheme. Applying the same idea to $\frac{d^3u}{dx^3} = 0$ problem I derived the scheme: $\begin{bmatrix} 1 & -3 & 3 & -1\end{bmatrix}$. Using said discretisation and adding Dirichlet constraints does not result in a singular system anymore. Nevertheless explicit schemes (e.g. Richardson, Jacobi, etc.) do not seem to converge for the resulting unsymmetric problem:

$$Ax = b$$

Instead I decided to symmetrize the problem by:

$$AA^Ty = b, \, x = A^Ty.$$

If CG is applied to the above form with some modification one can arrive at the CGNE algorithm. Using CGNE resulted in a convergent iteration.