Introduction

Numerical Methods for Differential Equations

Introduction

• 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. ∇•q = f (Heat transfer)

q = -K∇T (Fourier) Could be either constant or variable

e.g. deformation of elastic rod

Boundary conditions:

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 = k(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)

Initial condition

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

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

Inelastic problem example: dirichlet → imposed displacement neumann → imposed traction