Tutorial: Adjoint PDE for finding unstable equilibrium solutions of Navier-Stokes Equations

Introduction

This tutorial aims to illustrate the application of the adjoint-based approach for finding invariant solutions of the Navier-Stokes equations that was developed in Ref. [1]. The accompanying MATLAB codes may be used with proper citation to Ref. [1].

Overview of the method

Consider a set of partial differential equations (PDEs)

$\partial_t u=F(u),\quad\quad (1)$

where $F$ is a (nonlinear) differential operator acting on the function $u(x,t)$ . We assume that the differential operator $F$ only contains partial derivatives with respect to the spatial variable $x$ . We are interested in finding (relative) equilibrium solutions of PDE (1). These equilibria are typically unstable.

The goal is to derive a second PDE

$\partial_\tau u = G(u),\quad\quad (2)$

whose equilibria are stable and contain those of PDE (1) and further the solutions $u(\tau)$ of PDE (2) converge asymptotically to its equilibria for almost all initial conditions $u(0)$ . Note that the parameter $\tau$ is a fictitious time and does not in general coincide with time $t$ .

It is shown in Ref. [2] in the context of nonlinear waves and in Ref. [1] for the incompressible Navier-Stokes equation that the choice $G(u) = -\mathcal L^\dagger (u;F(u))$ accomplishes the above-mentioned goals. That is the solutions of

$\partial_\tau u = -\mathcal L^\dagger (u;F(u))$

converge asymptotically to the equilibria of Eq. (1). We refer to the above equation as the adjoint PDE.

Here the operator $\mathcal L$ is the Gateaux derivative (or linearization) of $F$ and $\mathcal L^\dagger$ is its adjoint with respect to some appropriate inner product.

Example

Consider the Kuramoto-Sivashinsky equation

$\partial_t u =- u\,\partial_x u-\partial^2_x u-\partial_x^4u,\quad x\in[0,L],$

with periodic boundary conditions.

The domain size $L$ plays the role of the “Reynolds number” here. It is known that for a large enough domain size $L$ , the invariant solutions of Kuramoto-Sivashinsky equation become unstable and the dynamics becomes chaotic. Ref. [3] presents a thorough study of the state space of the Kuramoto-Sivashinsky equation for $L=22$ . For the rest of this tutorial we set $L=22$ . The movie below shows a solution of the Kuramoto-Sivashinsky equation starting from a random phase initial condition.

For the Kuramoto-Sivashinsky equation, the nonlinear differential operator $F$ is given by

$F(u)=- u\,\partial_x u-\partial^2_x u-\partial_x^4u,$

whose Gateaux derivative reads

$\mathcal L(u;u')=-u'\partial_x u -u\partial_xu'-\partial_x^2u'-\partial_x^4u'.$

The adjoint of $\mathcal L$ with respect to the $L^2$ inner product is

$\mathcal L^\dagger(u;u'')=u\partial_x u''-\partial_x^2u''-\partial_x^4u''.$

One can readily check that

$\langle \mathcal L(u;u'),u''\rangle_{L^2} = \langle u',\mathcal L^\dagger(u;u'')\rangle_{L^2},\quad\quad \forall u,u',u''\in L^2(0,L).$

Therefore, the adjoint descent PDE reads

$\partial_\tau u = -\mathcal L^\dagger (u;F(u))$

We numerically solve this adjoint PDE with initial conditions $u(x,0)=A\sin(2m\pi x/L),\; m=1,2,3,\cdots$ In the following, we set $A=2$ .

Ref. [3] shows that the Kuramoto-Sivashinsky equation has three non-trivial equilibrium solutions for $L=22$ . These equilibria are labeled as E1, E2 and E3 in Figure 3 of Ref. [3]. We will use the same labeling here and find these equilibrium solutions by numerically integrating the adjoint PDE.

The movie below shows equilibrium E1 obtained from the initial condition $u(x,0)=2\sin(2\pi x/L)$ . The red dashed curve marks the initial condition and the black curve shows the evolution of the solution $u(x,\tau)$ of the adjoint PDE. The panel on the right shows the evolution of the $L^2$ residue as the fictitious time $\tau$ increases.

Note: The solution $u(x,\tau)$ of the adjoint PDE converges very quickly to the vicinity of the equilibrium E1 such that, after approximately the first 20 fictitious time units, it is visually indistinguishable from the equilibrium itself.

The movie below shows equilibrium E2 obtained from the initial condition $u(x,0)=2\sin(4\pi x/L)$ . The red dashed curve marks the initial condition and the black curve shows the evolution of the solution $u(x,\tau)$ of the adjoint PDE. The panel on the right shows the evolution of the $L^2$ residue as the fictitious time $\tau$ increases.

The movie below shows equilibrium E3 obtained from the initial condition $u(x,0)=2\sin(6\pi x/L)$ . The red dashed curve marks the initial condition and the black curve shows the evolution of the solution $u(x,\tau)$ of the adjoint PDE. The panel on the right shows the evolution of the $L^2$ residue as the fictitious time $\tau$ increases.

We found that for initial conditions with $m\geq 4$ , the solutions of the adjoint PDE converge to the trivial solution $u\equiv 0$ . The movie below shows an example with $m=4$ , i.e., $u(x,0)=2\sin(8\pi x/L)$

MATLAB code

This MATLAB code may be used with proper citation to Ref. [1].

References

Farazmand, M., An adjoint-based approach for finding invariant solutions of Navier-Stokes equations, J. Fluid Mech., vol. 795, pp. 278-312, 2016
Yang, J. & Lakoba, T. I., Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations. Studies in Applied Mathematics 118 (2), 153–197, 2007
Cvitanovic, P., Ruslan L. D., and Evangelos S., On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain., SIAM Journal on Applied Dynamical Systems 9.1, 1-33, 2010