Introduction

Numerical Methods for Differential Equations

Introduction

PDEs

• Elliptic
• Parabolic
• Hyperbolic
• Advection - Diffusion
• Stokes - Navier-Stokes

Unknown: u(x) u(x,t)

1) Elliptic Problems

\[-Δu = f\] or \[ -div(▽u) = f \] , or \[ \frac{∂^2u}{∂x^2} - \frac{∂^2u}{∂y^2} = f (\text{in 2D}) \]

e.g. \[ \int div \ q = f \quad (\text{Heat transfer}) \] \[ q = -k ▽T \ (\text{Fourier}) \] Could be either constant or variable e.g. deformation of elastic rod → boundary conditions

In order to solve the problem bc must be added:

• Γ_D: dirichlet boundary → b.c.: u = g(x)
• Γ_N: neumann boundary → b.c.: -▽u•n = h(x)
• Γ_R: robin boundary → b.c. linear combination of Γ_N, Γ_D: αu + β▽u • n = l(x)

Velocity of propagation of info is infinite in any direction: any perturbation is immediately felt everywhere.

2) Parabolic Problems

\[ \frac{∂u}{∂t} - Δu = f \] (Unsteady counterpart of Elliptic Problems)

1^st order time derivative requires an i.c.: u(x,0) = u₀(x) bc are the same as before, but g, h, l can be also functions of time

In elastic problem example:

• dirichlet → imposed displacement
• neumann → imposed traction

Numerical Methods for Differential Eqns

Introduction

PDEs model problems

• Elliptic
• Parabolic
• Hyperbolic
• Advection-Diffusion
• Stokes - Navier-Stokes

Unknown: u(x)

u(x, t)

1. Elliptic Problems

-Δu = f

or -div(∇u) = f , or ∂²u/∂x² + ∂²u/∂y² = f (in 2D)

e.g. ∫div q = f (Heat Transfer)

q = -k∇T (Fourier)

e.g. deformation of elastic rod

In order to solve the problem b.c. must be added:

• Γ_D: dirichlet boundary ➔ b.c.: u = g(x)
• Γ_N: neumann boundary ➔ b.c.: -∇u·n = h(x)
• Γ_R: robin boundary ➔ b.c. Linear combination of Γ_N, Γ_D:

αu +β∇u·n = ℓ(x)

Velocity of propagation of info is infinite in any direction: any perturbation is immediately felt everywhere

2. Parabolic Problems

∂u/∂t - Δu = f

(Unsteady counterpart of Elliptic Problems)

1^st order time derivative requires an i.c. : u(x,0) = u₀(x)

b.c. are the same as before, but g, h, ℓ can be also functions of time

• In elastic problem example: dirichlet ➔ imposed displacement
• neumann ➔ imposed traction

If purely Neumann conditions on all the boundary we have a constant & NOT well posed problem , so we need a piece of boundary with Dirichlet conditions.

In theory, in parabolic problems time can go to infinity (coil to interval); more often time domain of definition of the problem is limited, so :

Domain of definition of the problem: Ω x (0, T]

Velocity of propagation as in elliptic, is infinite ; there is, obviously, a preferential direction in time (forward, from i.c.), although backward problems in time are also common.

B) Hyperbolic Problems

• They include waves equation (sound, elastic...), Euler equations,...

We'll limit to 1^st order. Linear hyperbolic equations in 1D :

∂u/∂t + ∂u/∂x = 0

(Transport equation)

(Transport field) (velocity, for example)

• Since we do not have diffusion, the i.c. will be transported without deformations.
• For b.c. situation is totally ≠ from the previous ones, because we have a finite velocity of propagation of the info and a preferential direction : what happens downstream is influenced by what happens upstream.
• Info travels downstream → b.c. have to be provided upstream, not needed downstream

In multi-dimensional case:

Usually I have to provide b.c. in case I have ∂ · n < 0

(Domain of dependence: what happens downstream is influenced by upstream.)

• If a = const. analytical solution is u(x, t) = u₀ (x - a t)
• prediction of amplitude propagation...
• There are cases in which u is a vector, A a matrix ⇒ system of hyperbolic equations

∂u/∂t + A ∂u/∂x = 0

4) Advection-Diffusion Problems

Could be seen as 1.a. or 2.a. from mathematical point of view fits into elliptic/parabolic

1.a) -μ ▽ ⋅ ▽ u + b ⋅ ▽ u = f

(steady) (diffusion coefficient) (transport term)

2.a) ∂u/∂t -μ ▽ ⋅ ▽ u + b ⋅ ▽ u = f

More generally

_∂u/_∂t - μ Δu + b · ∇u + ru = f

• diffusion + transport - reaction
• advection - convection - reaction

• reaction (e.g. chemical reaction)

• If transport dominant, more similar to hyperbolic problems (|b| >> μ)
• If diffusion dominant, more similar to elliptic/parabolic equation (|b| << μ)

b.c. must be provided for all the domain (diff. term is the most important), and i.c. as well

Navier-Stokes or Stokes Equations

_∂y/_∂t + (y · ∇)y - μ Δy + ∇p = f

div (y) = 0

• non-linear term, one of the main issues
• Wide possibility of b.c.

b. Stokes problem (steady):

{ - Δu + ∇p = f

{ - div(u) = 0

meaningful approximation of Navier-Stokes, to understand its main issues

Properties of numerical methods we'll use to analyze problems

Generic physical problem

P(u,g) = 0

• unknown (scalar or vector)
• d: differential operator
• data

Well Posedness

• the solution exists
• the solution is unique
• the solution depends continuously on the data

∀_E>0 ∃_δ>0 ∃J⊆J(_E) ; P(u_δ, g_δ + δ_g) = 0 , |δ_j|∈J <= δ ⇒ |u-u_δ| <= _E

If bounded perturbation on the data; perturbation on the solution can be controlled ⟺ ‖u‖ ≤ ‖g‖_C

→ here a slight modification on the data implies great modifications on the solution (from u to 2 or more) ⇒ Not well-posed

We will always consider well-posed problems