(This content is originally written by ‪Kyongmin Yeo’s manual)

Introduction

The spectral method is solving certain differential equation by some “basis function”, typically sinusoids with Fourier method. With the Navier-Stokes equation, it can remove presssure term in N-S equation and solve viscous term analytically.

Pros:

  • Removing pressure term is huge performance advantage
  • Accurate result because differential operator doesn’t depends on grid size

Cons:

  • Only can be applied to periodic domain

Governing Equation

Original Navier-Stokes equation in convection form is

\[\begin{align} \dfrac{\partial u_i}{\partial t} &= -\dfrac{\nabla p}{\rho} - (u \cdot \nabla) u + \nu \nabla^2 u \\ \nabla \cdot u &= 0 \end{align}\]

Using following vector identity,

\[\begin{align} \dfrac{1}{2} \nabla (A \cdot A) = (A \cdot \nabla) A + A \times (\nabla \times A) \end{align}\]

The Navier-Stoke sequations in rotational form can be obatained. The reason is explained in the paper, Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations.

The reason is that pseudospectral approximation to the rotation, rather than Reynolds stress, form of the nonlinear terms of the Navier~Stokes equations semiconserves (cf. (Orszag 1971), Numerical simulation of incompressible flows within simple boundaries: Accuracy, Section 3) energy so that aliasing errors, although present, can not directly cause unconditional nonlinear instability

(Orszag 1971)
\[\begin{align} \dfrac{\partial u_i}{\partial t} &= -\dfrac{\partial P}{\partial x_i} + H_i + \nu \nabla^2 u \\ \dfrac{\partial u_i}{\partial x_i} &= 0 \end{align}\]

where

\[\begin{align} P &= \dfrac{p}{\rho} + \dfrac{1}{2} u_j u_j \\ H_i &= \epsilon_{i,j,k} u_j \omega_k = u \times (\nabla \times u) \end{align}\]

Removing pressure term

The pressure Poisson equation can be obatained by taking divergence from Navier-Stokes equation in rotational form

\[\begin{align} \nabla^2 P = \dfrac{\partial H_j}{\partial x_j} \end{align}\]

Expanding N-S equation and Poisson equation to Fourier space gives

\[\begin{align} \dfrac{d \hat{u}_i }{d t} &= -i \kappa_i \hat{P} + \hat{H}_i - \nu \kappa^2 \hat{u}_i \\ -\kappa^2 \hat{P} &= i \kappa_j \hat{H}_j \end{align}\]

Combining two equation and then

\[\begin{align} \dfrac{d \hat{u}_i }{d t} &= -i \kappa_i \left( -i \dfrac{\kappa_j}{\kappa^2} \hat{H}_j \right ) + \hat{H}_i - \nu \kappa^2 \hat{u}_i \\ \dfrac{d \hat{u}_i }{d t} &= -\dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i - \nu \kappa^2 \hat{u}_i \end{align}\]

where \( \kappa \) is a wavenumber. Final Navier Stokes equation is obtained without pressure term

\[\begin{align} \dfrac{d \hat{u}_i }{d t} &= -\dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i - \nu \kappa^2 \hat{u}_i \end{align}\]

Treating viscous term analytically

To treat a viscous terms analytically, multiply following formula to Navier Stokes equation w/o pressure form

\[f(t) = e^{\nu \kappa^2 t}\]

Then the equation becomes..

\[\begin{align} \left[ \dfrac{d \hat{u}}{dt} + \nu \kappa^2 \hat{u}_j \right] f(t) &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \\ f(t) \dfrac{d \hat{u}}{dt} + (\nu \kappa^2 f(t))\hat{u}_j &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \\ f(t) \dfrac{d \hat{u}}{dt} + (\nu \kappa^2 e^{\nu \kappa^2 t})\hat{u}_j &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \\ f(t) \dfrac{d \hat{u}}{dt} + \left(\dfrac{d e^{\nu \kappa^2 t}}{dt}\right)\hat{u}_j &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \\ \dfrac{d \hat{u}_i f(t)}{dt} &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \end{align}\]

this can be more simpler by introducing new term \(\widehat{NL}\)

\[\begin{equation} \dfrac{d \hat{u}_i e^{\nu \kappa^2 t}}{dt} = \widehat{NL} e^{\nu \kappa^2 t} \end{equation}\]

Time Discretization by RK3 method

For low-storage RK3 method (2-register, 3-stage, 3rd order), the coefficients are given by following table (Lundbladh et al. 1999), (Yu, Tsai, and Hsieh 1992), (Wray 1990)

order \(a_n\) \(b_n\) \(c_n\)
1st 8/15 0 0
2nd 5/12 -17/60 8/15
3rd 3/4 -5/12 2/3

Assume equations are given by following form,

\[\dfrac{\partial Q}{\partial t} = R(Q)\]

The low-storage RK3 method applied to the above equation using RK3 coefficients.

\[\begin{align} Q^1 &= Q^n + \Delta t \left( \dfrac{8}{15} R^n \right) \\ Q^2 &= Q^1 + \Delta t \left( \dfrac{5}{12} R^n - \dfrac{17}{60} R^1\right) \\ Q^{n+1} &= Q^2 + \Delta t \left( \dfrac{3}{4} R^n - \dfrac{5}{12} R^2\right) \end{align}\]

Before applying RK3 method to Navier-Stokes equation, apply low-storage RK3 method to reaction-diffusion equation

\[\begin{align} \dfrac{\partial \psi}{\partial t} &= G + L \psi \\ \psi^{n+1} &= \psi^{n} + a_n \Delta t G^n + b_n \Delta t G^{n-1} + (a_n + b_n) \Delta t \left(\dfrac{L \psi^{n+1} + L \psi^n}{2} \right) \end{align}\]

Then the Navier Stokes equation w/o pressure term can be discretized by above method

\[\dfrac{d \hat{u} e^{\nu \kappa^2 t}}{dt} = \widehat{NL} e^{\nu \kappa^2 t}\]

Discretization of LHS

\[\begin{align} LHS = \dfrac{\hat{u}^{n+1}_i e^{\nu \kappa^2 (t+a_n \Delta t + b_n \Delta t)} - \hat{u}^{n}_i e^{\nu \kappa^2 t}} {\Delta t} \end{align}\]

Discretization of RHS (denoting RHS as \( G \))

\[\begin{aligned} RHS &= \widehat{NL}^n e^{\nu \kappa^2 t} \\ &= a_n\widehat{NL}^n e^{\nu \kappa^2 t} + b_n\widehat{NL}^{n - 1} e^{\nu \kappa^2 (t - a_{n-1} \Delta t - b_{n-1} \Delta t)} \end{aligned}\]

Compensating \( e^{\nu \kappa^2 t} \) on both sides

\[\begin{align} \hat{u}^{n+1}_i e^{\nu \kappa^2 (a_n \Delta t + b_n \Delta t)} - \hat{u}^{n}_i = \begin{aligned}[t] & a_n \Delta t \widehat{NL}^n \\ &+ b_n \Delta t \widehat{NL}^{n - 1} e^{\nu \kappa^2 (- a_{n-1} \Delta t - b\_{n-1} \Delta t)} \end{aligned} \end{align}\]

Finally

\[\begin{align} \hat{u}^{n+1}_i = \begin{aligned}[t] &\left[ a_n \Delta t \widehat{NL}^n + \hat{u}^{n}_i \right ] e^{-\nu \kappa^2 (a_n + b_n) \Delta t} \\ & + b_n \Delta t \widehat{NL}^{n - 1} e^{-\nu \kappa^2 (a_n + b_n + a_{n-1}+ b_{n-1}) \Delta t} \end{aligned} \end{align}\]

or

\[\begin{align} \hat{u}^{n+1}_i &= \begin{aligned}[t] &\left[ a_n \Delta t \widehat{NL}^n + \hat{u}^{n}_i \right ] e^{-\nu \kappa^2 (c_n - c_{n+1}) \Delta t} \\ & + b_n \Delta t \widehat{NL}^{n - 1} e^{-\nu \kappa^2 (c_{n-1} - c_{n+1}) \Delta t} \end{aligned} \end{align}\]

Reference

  1. Orszag, Steven A. 1971. “Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations.” Studies in Applied Mathematics 50 (4): 293–327. https://doi.org/10.1002/sapm1971504293.
  2. Orszag, Steven A. 1971. “Numerical Simulation of Incompressible Flows within Simple Boundaries: Accuracy.” Journal of Fluid Mechanics 49 (1): 75–112. https://doi.org/10.1017/S0022112071001940.
  3. Lundbladh, A, S Berlin, M Skote, and C Hildings. 1999. “An Efficient Spectral Method for Simulation of Incompressible Flow over a Flat Plate.” TRITA-MEK Technical {\Ldots}.
  4. Yu, S T, Y L P Tsai, and K C Hsieh. 1992. “Runge-Kutta Methods Combined with Compact Difference Schemes for the Unsteady Euler Equations.” NASA Technical Memorandum, January. http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930006613.pdf.
  5. Wray, A A. 1990. “Minimal Storage Time Advancement Schemes for Spectral Methods,” January.