University:
Boston University
Course:
CAS MA 511 | Introduction to Analysis I
Academic year:

2023
Views:

85

Pages:

15
Author:

Annabel Combs

Extending to nonlinear systems q ⇤ t q` qr q ⇤ 1 1 q` = ↵ r A(q ⇤ A(q ⇤ q` ) = 1 A(q ⇤ q` ) = 1 f (q) = Aq ! 1 1 1 ) (↵ r (q ⇤ q` ) = ↵ Ar ⇤ f (q ) Rankine-Hugoniot condition for the constant coefficent linear system q` ) ` f (q ) = 1 (q ⇤ q` ) Rankine-Hugoniot conditions For a 2x2 linear system, we have A(q ⇤ A(qr q` ) = ⇤ q )= 1 2 (q ⇤ q` ) (qr ⇤ q ) For f (q) = Aq , we can write this as : The left and right states q` and qr as “connected” by an ⇤ q intermediate state . Constant coefficient Riemann problem q t q` ? ⇤ qr ⇤ We could have asked “Find an intermediate state q such that ” For f (q) = Aq , this leads to the eigenvalue problem that we solved. Extending to nonlinear systems t q` qr Reminder : Solutions to the constant coefficient linear system travel along characteristic curves : must be an eigenvalue of , i.e. Solution remains constant along straight lines Nonlinear shallow water wave equations qt + f (q)x = 0 where q= ✓ h hu ◆ , f (q) = ✓ hu 1 2 2 hu + 2 gh ◆ for smooth solutions, this can also be written as where 0 f (q) = ✓ is the flux Jacobian matrix. 0 2 u + gh 1 2u ◆ Show! What changes in the nonlinear case? qt + f (q)x = 0 We can still ask “Are there characteristic curves on which the solution remains constant? For smooth solutions, we have where is the flux Jacobian matrix. Characteristics are governed by eigenvalues of the flux Jacobian Shallow water wave equations q= ✓ ◆ ✓ hu , f (q) = 1 2 hu + 2 gh2 ✓ ◆ 0 1 0 f (q) = u2 + gh 2u h hu ◆ Eigenvalues and eigenvectors of the flux Jacobian : p p 1 2 =u gh, = u + gh ✓ ✓ ◆ ◆ 1p 1p 1 2 r = , r = u u + gh gh Eigenvalues and eigenvectors depend on q! What can happen? ? q` t qr 1-characteristics ? q` 2-characteristics qr Riemann solution for the SWE Consider the case where : Assume that left and right states are constant in this infinitesimal box Using the conservation law, we can write Show! Riemann problem for SWE This leads to This is the required jump condition across shocks. generally we can write this condition as More Rankine Hugoniot jump condition where is the shock speed. q` qr Rarefaction waves Consider the case where Let for : be the slope of the characteristic. We need to find . Recall that . Then slope = The denominator is never 0! q` qr Solve resulting ODE to get in the centered rarefaction. Centered rarefaction Solve the system of two ODEs (for SWE) : subject to Use Riemann invariants to solve for unknown constants. q` qr Solving the Riemann problem q t q` ? ⇤ qr Find a state such that is connected to by a physically correct 1-shock wave or 1-rarefaction, and is connected to by a physically correct 2-shock or 2-rarefaction. The need to find a state that simultaneously satisfies both conditions above means we have to solve something... Solving the Riemann problem 1 k oc 2-s h n o i t c a -raref Formulas for these curves are relatively simple. oc sh 1k 2-rarefaction Curves represent states that can be connected to or by a shock or a rarefaction. Use a nonlinear root-finder to find the middle state . • Determine the structure of the rarefaction (if there is one). Riemann solution 1-rarefaction t q` q ⇤ 2-shock qr The structure of the Riemann solution depends on the initial conditions.