Fourier spectral method

Together with Johanna Ulvedal Marstrander, we have uploaded an arXiv preprint concerning the justification of the Fourier spectral method for quasilinear first-order systems of balance laws.

The Fourier spectral method¹ is a strategy for the spatial discretization of evolution equations (on the torus or on the full space and assuming sufficient decay at infinity) which takes advantage of the Fast Fourier Transform to efficiently approximate operations such as pointwise multiplication and composition as well as Fourier multiplication. Fourier multiplication is a pointwise operation on Fourier coefficients, and includes in particular differential operators. Such strategy is quite natural to apply on water waves models, and even more natural for fully dispersive systems² systems since they involve Fourier multiplications. They are at the center of the numerical package WaterWaves1D.jl that we developped with Pierre Navaro³.

Our aim with Johanna was to rigorously justify the Fourier spectral method for spatial-semi-discretization. Applied to the Saint-Venant (shallow-water) system $$ \begin{cases} \partial_t \eta+\nabla\cdot ((1+\eta) {\bf u}) =0, \\ \partial_t {\bf u}+\nabla\eta + ({\bf u}\cdot\nabla {\bf u}) =0, \end{cases} $$ the question we ask is the following: can we prove the (spectral, i.e. with an exponential decay rate with respect to $N$) convergence $(\eta_N,{\bf u}_N)\to (\eta,{\bf u})$ as $N\to\infty$ where $(\eta_N,{\bf u}_N)$ are the solutions to $$ \begin{cases} \partial_t \eta+\Pi_N\circ\Big\{\nabla\cdot ((1+\eta) {\bf u})\Big\} =0, \\ \partial_t {\bf u}+\Pi_N\circ\Big\{\nabla\eta + ({\bf u}\cdot\nabla {\bf u})\Big\} =0, \\ (\eta_N,{\bf u}_N)(t=0,\cdot)=(\Pi_N\circ\eta,\Pi_N\circ{\bf u})(t=0,\cdot) \end{cases} $$ where $ \Pi_N $ is the low-pass filter defined as a Fourier frequencies cut-off: $$ \widehat{\Pi_N\circ f}({\bf k})= \begin{cases} \widehat{ f}({\bf k})& \text{if } |{\bf k}|\leq N,\\ 0& \text{otherwise.} \end{cases}$$ (here, if ${\bf k}$ is a vector, $|{\bf k}|$ is the sum of absolute values of its entries).

It may come to a surprise that we were not able to answer positively this question in full generality. The issue is that because the symbol of the operator $\Pi_N$ is a sharp cut-off, it misses the smoothness that allows for symbolic calculations. In a nutshell, it is not true that the linear operator $g\in L^2(\mathbb R^d) \mapsto \Pi_N\circ \big(f \, ({\rm Id}-\Pi_N)\circ \nabla g\big)\in L^2(\mathbb R^d)$ is bounded uniformly with respect to $N$, no matter how smooth the function $f$ is. Yet such a property is used in the energy method which is suitable for quasilinear systems such as the Saint-Venant system. A consequence is that, in theory, one can expect that the semi-discretized approximation $(\eta_N,{\bf u}_N)$ may suffer from alterations (such as high-frequency amplifications) preventing the desired spectral convergence.

This issue being unveiled, one is led to consider two strategies:

1. using low-pass filters with smooth (Lipschitz suffices) symbols instead of the crude frequency cut-off;
2. using additional structural assumptions on the system at stake (we consider Hamiltonian structure).

We discuss in details both situations and provide sufficient conditions for spectral convergence.

Our study is accompanied with some numerical investigations performed (and fully reproducible) with numerical package WaterWaves1D.jl. Our aim was to exhibit some situations where the use of smooth low-pass filters instead of the commonly used crude frequency cut-off was mandatory. Despite our efforts we have not been able to satisfactorily provide such an example within the framework of our study⁴. The reason for such unexpected “good” behavior of the crude frequency cut-off is a question left open.