Spectral method: Difference between revisions

Revision as of 22:22, 28 August 2022

Revision as of 22:23, 28 August 2022

Line 58:

where ρ is the [[viscosity]] coefficient. In weak conservative form this becomes

:<math>\left\langle \partial_{t} u , v \right\rangle = \left\langle \partial_x \left(-\frac{1}{2} u^2 + \rho \partial_{x} u\right) , v \right\rangle + \left\langle f, v \right\rangle \quad \forall v\in \mathcal{V}, \forall t>0</math>

where \overline{g(x)}\,dx</math> following [[inner product space|inner product]] notation. [[integration by parts|Integrating by parts]] and using periodicity grants

where <math>\langle f, g \rangle := \int_{0}^{2\pi} f(x)

\overline{g(x)}\,dx</math> following [[inner product space|inner product]] notation. [[integration by parts|Integrating by parts]] and using periodicity grants

:<math>\langle \partial_{t} u , v \rangle = \left\langle \frac{1}{2} u^2 - \rho \partial_{x} u , \partial_x v\right\rangle+\left\langle f, v \right\rangle \quad \forall v\in \mathcal{V}, \forall t>0.</math>

To apply the Fourier-Fourier–[[Galerkin method]], choose both

:<math>\mathcal{U}^N := \left\{ u : u(x,t)=\sum_{k=-N/2}^{N/2-1} \hat{u}_{k}(t) e^{i k x}\right\}</math>

and