INTRODUCTION TO PDEs

In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototypeboundary value problems (BVP) and initial boundary value problems (IBVP) to study the methods. We begin by giving a brief overview of PDEsand introducing some terminology. We will concentrate mainly on second order PDEs.

Note that a second order PDE can be classified as one of 3 types: elliptic, parabolic, or hyperbolic.

1 BVPs for elliptic equations

Suppose we are given an open, bounded, connected domain in R2 denoted by Ω with smooth boundary ∂Ω. The prototype problem for second order

elliptic problems is:

Seek a (real-valued) function u = u(x, y) defined on Ω = Ω ∪ ∂Ω satisfying

−∆u = f(x, y) (1)

where ∆w = wxx + wyy with wxx denoting the second partial derivative of w with respect to x. (1) is called Poisson’s equation, ∆ is called theLaplacian operator and (1) is called Laplace’s equation in the special case f = 0. In general, we say the DE is homogeneous if the source term is zero.

Note that in R3 the Laplacian operator is just ∆w = wxx + wyy + wzz

For a specific f(x, y) we know that there are many such u’s satisfying (1). To uniquely specify a u we must impose auxiliary conditions, in this caseboundary values for u or its derivatives, or a combination of both. (We will see that we need to be careful here!)

If we specifyu(x, y) = g(x, y) ∀(x, y) ∈ ∂Ω (2)

then we are imposing a Dirichlet boundary condition . If g = 0, we say it is a homogeneous Dirichlet boundary condition; otherwise we call theboundary condition inhomogeneous.

Equations (1) and (2) form a Dirichlet BVP for u.

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If we specify∂u

∂~n= g(x, y) ∀(x, y) ∈ ∂Ω (3)

where ~n is the unit outer normal and∂u

∂~n= ∇u · ~n. Equations (1) and (3) form a Neumann BVP.

Remark: The Neumann BVP is not uniquely determined since if u satisfies Equations (1) and (3) then so does u + C where C is an arbitraryconstant.

We can also specify a mixed or Robin boundary condition of the type

αu(x, y) + β∂u

∂~n(x, y) = g(x, y) (4)

Here, if α = 0 then we have a Neumann boundary condition and if β = 0 we have a purely Dirichlet boundary condition.

Note that we can write ∂Ω = ∂Ω1 ∪ ∂Ω2 where ∂Ω1 ∩ ∂Ω2 = 0 and specify one type of boundary condition on ∂Ω1 and another on ∂Ω2.

Remark: For sufficiently smooth boundary (say piecewise C1), f continuous in Ω and g continuous on ∂Ω, the Dircihlet problem (1) -(2) has aunique solution. For cases where Ω is simple (such as a rectangle, circle, etc), we can use the technique of separation of variables to obtain an infiniteseries form of the exact solution.

Example: Consider Poisson’s equation with homogeneous Dirichlet boundary data posed on the unit square:

−∆u = f(x, y) = x2y in Ω = (0, 1) × (0, 1)

u = 0 on ∂Ω

whose exact solution is given by

u(x, y) =

∞∑

n,m=1

γm,n sin(nπx) sin(mπy)

with

γn,m =4

n2(m2 + n2)

∫

1

0

∫

1

0

f(x, y) sin(nπx) sin(mπy) dxdy

When the domain is more general or we have variable coefficients for example, then we must turn to approximating the solution to the BVP. In fact,even though we have the exact solution to the above example, it is in terms of an infinite series and convergence is slow. In addition, we may have touse numerical integration to evaluate the coefficients γn,m for certain choices of f(x, y).

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Physical setting:

Elliptic problems model states of equilibrium. For example, consider a membrane whose boundary is attached to a fixed frame. u is the displacementat (x, y, t). For elliptic equations we consider only the equilibrium shape of the membrane. We give a force f independent of t and displacement u

at ∂Ω. We seek u satisfying the given displacement on the boundary such that −∆u = f . This equation often arises in electric potential theory,equilibrium flow, irrotational 2-D steady motion of an inviscid incompressible fluid.

2 IVP and IBVP for parabolic and hyperbolic equations

Parabolic and hyperbolic equations model the time evolution of a physical phenomena from some know time (such as t = 0).

The prototype equation for parabolic type second order equations is the heat (or diffusion) equation. We can consider either a purely initial valueproblem or a initial boundary value problem. In one space dimension we have the following.

Seek u(x, t) satisfying the purely initial value problem

ut = νuxx −∞ < x < ∞, t > 0 (5)

u(x, 0) = u0(x) −∞ < x < ∞ (6)

where ν > 0 is a given constant and u0 is a given function of x. This is called the IVP (or Cauchy problem) for the heat equation in one dimension.

We can also consider an IBVP for the heat equation when we have a bounded domain, say [a, b]. Here we choose to impose Dirichlet boundary data.

Seek u(x, t) satisfying the IBVP

ut = νuxx a < x < b, t > 0 (7)

u(x, 0) = u0(x) a ≤ x ≤ b (8)

u(a, t) = f(t) u(b, t) = g(t) t ≥ 0 (9)

Of course, we must have that f(0) = u0(a) and g(0) = u0(b) for the problem to be well-defined. If the diffusion equation is defined on a domain inR

2 then we have the obvious extension. For example, for the IBVP defined on [a, b] × [c, d] with Dirichlet boundary data we have:

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Seek u(x, y, t) satisfying the IBVP

ut = ν∆u = ν(uxx + uyy) a < x < b, c < y < d t > 0

u(x, y, 0) = u0(x, y) a < x < b, c < y < d

u(a, t) = f(t) u(b, t) = g(t) t > 0

Physical setting: temperature diffusion, diffusion of some concentration

A second type of equation which models a time-dependent process is the hyperbolic type. The prototype equation for hyperbolics is the wave equation.An IBVP for the wave equation in one space dimension is given by the following.

Seek u(x, t) satisfying the IBVP

utt = c2uxx a < x < b, t > 0 (10)

u(x, 0) = u0(x), ut(x, 0) = v0(x) a ≤ x ≤ b (11)

u(a, t) = f(t) u(b, t) = g(t) t ≥ 0 (12)

Note that in this case we need to impose two initial conditions since the time derivative appears as a second derivative.

Of course we could also impose Neumann or mixed boundary conditions for these IBVPs.

Physical setting: A physical example of the wave equation is a string where u is the displacement.

One might ask how it is possible to determine if a particular second order linear PDE is elliptic, hyperbolic or parabolic. A general second orderlinear PDE in two variables (not necesarily spatial variables) can be written as

auxx + buxy + cuyy + dux + euy + fu = g

The discriminant, b2 − 4ac, determines the classification of the equation as follows

if b2 − 4ac

= 0 the equation is parabolic< 0 the equation is elliptic> 0 the equation is hyperbolic

For example, in Poisson’s equation, a = c = 1, b = 0, (d = e = f = 0 also) so b2 − 4ac = −4 < 0 so the equation is elliptic. For the heat equation wehave a = 1, b = c = 0 so b2 − 4ac = 0 and the equation is parabolic.

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3 Other classifications

All of the equations we have considered thus far are linear equations. A differential equation (DE) is linear if the unknown and its derivatives appearlylinearly. Otherwise the equation is nonlinear. .

A standard example (in one space dimension) of a second order nonlinear equation is Burger’s equation

ut + uux = νuxx (13)

Other examples include−∆u + g(u) = f(x, y)

where g(u) is a nonlinear function of u such as u3 or sin u.

We can also consider systems of differential equations. For example, equations modeling fluid flow are well known. The steady state Navier-Stokesequations are given by

−ν∆~u + ~u · grad ~u + grad p = f

div ~u = 0

plus appropriate boundary conditions.

Of particular interest to us are first order (symmetric) hyperbolic systems. For example, if ~u = (u1, u2, . . . un) satisfies

∂~u

∂t= A

∂~u

∂x

where A is an n × n matrix.

Remark: In the sequel we shall assume that the mathematical problem we want to approximate has a unique solution with sufficient continuityand differentiability properties (i.e, sufficient regularity ) and that the solution depends continuously on the data.

Remark: We will see that to show a particular numerical method results in an approximation which converges to the exact solution at an optimalrate, we will have to require stronger continuity and differentiability conditions on the solution than was required for the proof of existence anduniqueness. This is not unique to finite difference methods (FDM).

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