Hyperbolic Problems

Numerical Methods for Differential Eqns

Hyperbolic PDEs

Totally different nature with respect to others

Let's start our analysis from conservation laws:

\(\frac{\partial}{\partial t} \int_{\Omega} u \, d\Omega + \oint_{\partial \Omega} f(u) \, ds = 0\)

In general could be \(f(u) = a(u)\)

• \(\frac{\partial u}{\partial t} + \frac{\partial}{\partial x} (a(u)) = 0\) (Non conservative form)
• If \(a(u) = \text{const.} \Rightarrow \frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0\)
• If \(f(u) = \frac{1}{2} u^2\) → Burgers
• \(\frac{\partial u}{\partial t} + \frac{\partial}{\partial x}(\frac{1}{2} u^2) = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0\)
• Total derivative is zero!
• \(x(t) - x_0 = a(u) t \Rightarrow x(t) = x_0 + a(u) t\)

\(\int_{x_0}^{x} \frac{1}{a(u)} dx = t\) → Characteristic lines

\(u(x,0) = u_0(x)\) → unbounded, i.e., \(u(x,0) = u_0(x)\)

\(u(x,t) = u_0(x - at) \)

Periodic b.c.: trace back to the foot

Generic (non-periodic) b.c.: only on inflow

Inflow is where \(a \cdot \hat{n} < 0\)

Exist \(\Gamma\) such that \( A = \Gamma \Lambda \Gamma^{-1} \)

Where \(\Lambda\) is a diagonal matrix containing all eigenvalues of \(A\), eigenvalues being real numbers \(\Lambda = \text{diag}(\lambda_1, \ldots, \lambda_p)\), \(i \in \mathbb{R}\)

Condition for system to be hyperbolic is \(A\) can be reduced to diag through \(\Gamma\)

_∂u₁ + ΓΛΓ^-1 _∂u₁ = ΓΛΓ^-1 _∂

∂t _∂x

define w = Γ^-1u

_∂w + Λ _∂w = 0

∂t _∂x

⇒1 is a decoupled system if 4. ΓΛΓ^-1 holds

⇒New variables w are called characteristic variables

p characteristic lines

⇒Suppose p=3, p×p system

λ₁ 70

λ₂ 70

λ₃ 60

• for a system inflow and outflow must be referred to each variable of the system, not to the system as a whole.
• o is inflow for w₁, w₂, outflow for w₃
• i is inflow for w₃, outflow for w₁, w₂

• b.c. must be provided on proper points (inflow & variable)

In general, set of points x - λ_it is called the Domain of dependence of the solution

D(x,t) {x - λ_it, i = ..., p }

w = Γ^-1u ; A = ΓΛΓ^-1 → _∂w_i + λ_{i ∂}w_i

∂t _∂x

• If A is diag. Γ will be a linear combination, Γ and Γ^-1 will be just coefficients, related to eigenvectors of A
• If A(u) in Γ we have solution itself, and mapping from characteristic to real variables is more difficult

How do we solve hyperbolic problems?

Finite Volume

• → Very closely related to finite differences

• We would like the numerical method to recover somehow the conservation of U
• finite elements are not so good, classical Lagrangian finite elements are not conservative, not the best method to solve hyperbolic problems
• →We look for a numerical method which mimics as much as possible dynamics of the physics problem:

real → model

• elements in finite volume are called cells □ or ▽ or anything else
• cubes (quadrilaterals) or parallelepipeds (exahedra) are easier to dere.
• If very complicated you can have complex boundaries not so good
• In practice we even have generic elements □ from the boundary, then close ∅ the boundary more structured mesh → good grid around boundaries, erosion away

• We’ll stick to 1D case: λ=const.

∂u + λ ∂u = ∂u + ∂(λu)

∂t ∂x ∂t ∂x