Spectral method: Difference between revisions

Revision as of 22:07, 14 July 2022

Revision as of 22:22, 28 August 2022

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===A concrete, linear example===

Here we presume an understanding of basic multivariate [[calculus]] and [[Fourier series]]. If <math>g(x,y)</math> is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, <math>g(x,y)=g(x+2\pi,y)=g(x,y+2\pi)</math>) then we are interested in finding a function ''f''(''x'',''y'') so that

:<math>\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)f(x,y)=g(x,y)\quad \text{for all } x,y</math>

<!--math>f_{xx}(x,y)+f_{yy}(x,y)=g(x,y)\quad \text{for all} x,y</math-->

where the expression on the left denotes the second partial derivatives of ''f'' in ''x'' and ''y'', respectively. This is the [[Poisson equation]], and can be physically interpreted as some sort of heat conduction problem, or a problem in potential theory, among other possibilities.

If we write ''f'' and ''g'' in Fourier series:

:<math>f=:\sum a_{j,k}e^{i(jx+ky)}</math>

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We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that ''f'' has a continuous second derivative. By the uniqueness theorem for Fourier expansions, we must then equate the Fourier coefficients term by term, giving

{{NumBlk|:(*)| <math>a_{j,k}=-\frac{b_{j,k}}{j^2+k^2}</math>|{{EquationRef|<nowiki>*</nowiki>}}}}

which is an explicit formula for the Fourier coefficients ''a''_''j'',''k''.

With periodic boundary conditions, the [[Poisson equation]] possesses a solution only if ''b''_''0'',''0'' = 0. Therefore, we can freely choose ''0a''_0,0 which will be equal to the mean of the resolution. Therefore,This corresponds to choosing the integration constant.

we can freely choose ''a''_''0'',''0'' which will be equal to the mean of the resolution. This corresponds to choosing the

integration constant.

To turn this into an algorithm, only finitely many frequencies are solved for. This introduces an error which can be shown to be proportional to <math>h^n</math>, where <math>h := 1/n</math> and <math>n</math> is the highest frequency treated.

==== Algorithm ====

# Compute the Fourier transform (''b_j,k'') of ''g''.

# Compute the Fourier transform (''a_j,k'') of ''f'' via the formula ({{EquationNote|*}}).

# Compute ''f'' by taking an inverse Fourier transform of (''a_j,k'').