Consider a PDE of the form
$$ \frac{\partial}{\partial t} u(t,x) \enspace = \enspace F \Big( t, x, u, \tfrac{du}{dx}, \tfrac{d^2u}{dx^2} \Big) \quad , \qquad x \in [-1,1] \quad .$$
I want to solve this numerically by means of spectral methods.
To this end, I consider two expansions of $u$ (in terms of Lagrange polynomials $L_n(x)$ and in terms of Chebychev polynomials $T_n(x)$), such that
$$u(t,x) \enspace = \enspace \sum_{n=0}^N u_n(t) \cdot L_n(x) \enspace = \enspace \sum_{n=0}^N \lambda_n(t) \cdot T_n(x) \quad , $$
where $u_n(t) \equiv u(t,x_n)$ and where $\lambda_n(t)$ are the corresponding spectral coefficients. I have already established, that the coefficients for the derivative $\tfrac{du}{dx}$ (and also higher order derivatives) can be calculated in $\mathcal{O}( N \ln N )$ time by means of Fast Fourier Transformation, in detail the idea is:
- Transform into spectral space in $\mathcal{O}(N \ln N)$ time.
- Differentiate in $\mathcal{O}(N)$ time.
- Transform back into physical space in $\mathcal{O}( N \ln N )$ time.
This is some standard procedure in spectral methods. By this, given an initial condition $u(0,x)$, one can evolve the above PDE via implementation of e.g. a Runge-Kutta-Scheme.
Now, if the PDE looks a bit different, namely
$$ \frac{\partial^2}{\partial t \partial x} u \enspace = \enspace F \Big( t, x, u, \tfrac{du}{dx}, \tfrac{d^2u}{dx^2} \Big) \quad , \qquad x \in [-1,1] \quad ,$$
this is more difficult. I am quite frankly stuck on how to approach this. Certainly, all derivatives can be calculated in $\mathcal{O}(N \ln N)$ time, however a Runge-Kutta-Scheme cannot be applied to evolve the equation, since the RHS is a mixed derivative. Moreover, the differentiation matrix is not invertible (which makes sense because the information about constant terms gets lost), therefore I cannot rewrite the PDE such that there is an explicit expression for $\tfrac{\partial u}{\partial t}$.
Does anyone know/has hints/is aware of literature on how to circumvent this problem?