I’m implementing a numerical solver to a certain kind of a 2D stochastic PDE using Fourier Transforms.
So, this solver is a Galerkin type method, which means that choosing the right basis functions I can apply Fourier Transforms to compute the solution.
Long story short, applying the explicit Euler-Maruyama scheme to my SPDE and discretising the domain with N^2 nodes, the discrete version of the Fourier coefficients of
V at time step
t_i+1 is given by:
where the hat terms corresponds to the fourier coefficients of the equation,
epsis the level of additive noise,
Wis a complex random matrix with distribution N(0,1) and
\lambdais a spatial correlation matrix.
Notice that the noise is being added in the frequency domain!
However, when comparing results with other numerical methods the effect of the noise is not the same, actually I have to scale up
eps by a factor.
I’m using the
fftw.jl package to compute the Fourier Transforms. Thus, it’s known that
fft(V) = N^2*v_mn, i.e. the function
fft gives us the fourier coefficients of a certain vector (matrix) scaled by a factor of
So, the stochastic term of my expression showed above has not the same magnitude as the rest of the coefficients. Am I right?
To overcome this issue, I simply have to multiply
N^2, is this easy? Or adding noise in the frequency domain is more complicated than this?
Actually, an experimental evidence that the number of nodes influences the noise effect in the solution, is if I increase
N I’ll also have to increase
eps to continue to have similar behaviours than with less nodes.
Sorry for my question, but I’m a bit newbie in the Fourier world.