Review of Fourier Analysis for PDE

Fourier analysis has been a powerful tool for many years in analyzing different numerical procedures, and it goes back to Von Neumann. It can be used for a variety of tasks including a quantitative information about the behavior of solutions, their dependence on boundary values, etc. It can also be used to analyze numerical procedures that we use to solve partial differential equations or optimization problems.

We go here briefly on the main ideas we need for the analysis of optimization problems. Let us begin by a simple problem, the linearized small disturbance potential equation of fluid dynamics. This equation will demonstrate the different issues we are concerned with, and will set the foundation for the treatment of general problems, including systems of equations such as the Euler and the Navier-Stokes equations. Example I: A Scalar Equation. Consider the equation

$$(1 - M^2 ) \phi _{xx} + \phi _{yy} + \phi _{zz} = 0,$$ (1)

in the domain $\Omega = \{ (x,y,z) \vert z > 0 \}$. This equation is an approximation for a flow in the x direction, where perturbation velocities (around a mean velocity) are related to the potential $\phi$ as $u = \phi _x, v , \phi _y , w = \phi _z$, see Hirsch [12].

We want to analyze the solution in terms of its boundary values on $z=0$. We consider boundary data in terms of Fourier components

$$\phi ({\bf x}, 0 ) = \exp ({i {\bf k} \cdot {\bf x} ) },$$ (2)

where ${\bf x} = (x,y)$ and ${\bf k} = (k_1, k_2)$. We look for an exponential behavior in the $z$ direction

$$\phi ({\bf x} ,z ) = \exp ({i {\bf k} \cdot {\bf x} )} \exp ({ \sigma z } ),$$ (3)

and a substitution of this expression into equation (1) for $\phi$ leads to the relation

$$\sigma ^2 = (1 - M^2) k_1 ^2 + k_2^2 .$$ (4)

Note that this equation for $\sigma$ has two solutions. For $M <1$ one of them, $\sigma _1 < 0$, correspond to a decaying solution for $\phi$ as a function of z and is considered as the solution of interest. The other one, $\sigma _2 > 0$, correspond to an unbounded solution for $\phi$ and is discarded. Thus, for the subsonic case we have

$$\phi ({\bf x}, z) = \exp ( i {\bf k} \cdot {\bf x} )\exp \left( - z \sqrt{ (1-M^2)k_1^2 + k_2^2 } \right)$$ (5)

For the supersonic case, $M>1$, there are two bounded solutions for $(1 - M^2) k_1 ^2 + k_2^2 < 0$, and only one bounded solution for $(1 - M^2) k_1 ^2 + k_2^2 \geq 0$. The two bounded solutions can be viewed as an incident wave and a reflected wave, whose sum satisfies the boundary condition,

$$\begin{array}{rl} \phi _i ({\bf x}, z) = A _i \exp ( i {\bf k} \c... ... \sqrt{ -(1-M^2)k_1^2 - k_2^2 } \right) & (1-M^2) k_1^2 + k_2^2 < 0 \end{array}$$ (6)

A general solution for the problem can be written as a sum (integral) of these Fourier components.

Example II: A System of PDE. We next show the use of Fourier analysis for studying systems of partial differential equation. Consider the system

$$\left( \begin{array}{cr} \beta^2 \frac{\partial }{\partial x}& \f... ...array}\right)= \left( \begin{array}{c} 0 \\ 0 \end{array}\right) \qquad \Omega$$ (7)

where $\Omega = \{ (x,y) \vert y \geq 0 \}$ and $\beta ^2 = 1 - M^2$. Here one considers vector Fourier components, that is

$$\left( \begin{array}{c} u \\ v \end{array}\right) = \left( \begin{array}{c} A \\ B \end{array}\right) \exp ( i k_1 x ) \exp ( \sigma y ).$$ (8)

The problem is to determine $\sigma$ but also the relation of $A$ and $B$. This is done by substituting this form into the equations leading to an algebraic equation for $A,B$,

$$\left( \begin{array}{cc} i \beta ^2 k_1 & \sigma \\ \sigma & - i... ... \\ B \end{array}\right) = \left( \begin{array}{c} 0 \\ 0 \end{array}\right).$$ (9)

This is a linear set of equations for the unknown $A,B$ and it has a non zero solution if and only if the determinant of the system is zero. This gives an equation for $\sigma$ in terms of $k_1$,

$$\beta ^2 k_1^2 - \sigma ^2 = 0.$$ (10)

For the subsonic case $\beta ^2 > 0$ and we have only one solution $\sigma = - \beta \vert k_1 \vert$, which correspond to a bounded solution in $\Omega$. To find the relation of $A$ and $B$ we substitute this value of $\sigma$ into the system, and solve for $A,B$ yielding,

$$\left( \begin{array}{c} A \\ B \end{array}\right) = \left( \begi... ...end{array}\right) \qquad \sigma = - \beta \vert k_1 \vert , \qquad \beta ^2 > 0$$ (11)

For the supersonic case, $\beta ^2 < 0$, we have two solutions $\sigma _+ = i \vert \beta \vert k_1$ and $\sigma _- = - i \vert \beta \vert k_1$ corresponding to two bounded solution in $\Omega$, and the coefficients $A,B$ are given by

$$\left( \begin{array}{c} A _\pm \\ B _\pm \end{array}\right) = \l... ...}\right) \qquad \sigma _\pm= \pm i \vert \beta \vert k_1 , \ \qquad \beta ^2 <0$$ (12)

The solution for $A,B$ given by (11) or (12) can be multiplied by an arbitrary constant. The actual value of that constant depends on the boundary condition which is required for the problem. This complete the solution of the problem in $\Omega$.

Remark: In general, there may be several solutions for $\sigma$ and to each of them there would correspond a vector solution such as (11) or (12) in the example above. The general solution is then a linear combination of these vectors with some weights to be determined from the boundary conditions. If the problem is well posed, then the number of boundary condition equals exactly to the number of bounded solutions that we are seeking and the analysis can be completed.

Some of the important question regarding general systems of equations can be answered to some extent using this tool. These include well-posedness of the problem, the choice of boundary conditions, their effect on the solutions and more.