Posted: October 6th, 2016

Integral curves Recall once more that for an m-dimensional sytem, the m right eigenvectors may be used to define m so-called integral curves v i (ξ), in v space and paramterised by ξ, according to v i (ξ) = α(ξ)r i (ξ) (1) and along which there are ** m − 1 distinct functions w i (ξ)** which are i-Riemann invariants and remain constant along the integral curve

Week 3: Simple waves In this notebook we introduce the concept of simple waves by considering the special case of rarefactions. 1.0.1 Integral curves Recall once more that for an m-dimensional sytem, the m right eigenvectors may be used to define m so-called integral curves v i (ξ), in v space and paramterised by ξ, according to v i (ξ) = α(ξ)r i (ξ) (1) and along which there are ** m − 1 distinct functions w i (ξ)** which are i-Riemann invariants and remain constant along the integral curve. Hence, d dξ w i (ξ) = 0, (2) ∇w i (ξ) · d dξ v i (ξ) = 0, (3) and, ∇w i (ξ) · d dξ r i (ξ) = 0. (4) 1.0.2 Simple waves Assume a smooth mapping exists of the form ξ(x, t) such that every point in x − t maps onto a point on a single integral curve v i (ξ). This is a simple wave. A consequence of this definition is that every i-Riemann invariant is constant throughout a simple wave. For a simple wave satisfying vt + fx = 0, (5) where f is a function of v, we must then have ξt + λ i ξx = 0. (6) 1 1.0.3 Rarefactions As a special case of simple waves we consider rarefactions. These are simple waves for which ξ(x, t) = x/t. Hence, using the equation ξt + λ i ξx = 0 from the previous cell, we obtain λ i = x t = ξ. (7) We assume the rarefaction smoothly connects the states on a single integral curve between the point at ξ1 and the point at ξ2. We take the state connected to the left in the x direction as vl ≡ v(ξ1) and the state to the right as vr ≡ v(ξ2). Hence, ξ1 = λ i (vl) ξ2 = λ i (vr). (8) Using ξ = λ i , we may obtain the expression (show this) d dξ v = r i ∇λi · r i (9) 1.1 Simple rarefaction in isentropic system Assume we seek a simple 1-rarefaction for the isentropic system where the left state at ξ = ξ1 is known and is given as [ρl ul ] T . We have v =  ρ u  λ 1 = u − c r1 =  − ρ c 1  w 1 = u + c ln ρ (10) Hence, ∇λ 1 =  0 1  (11) and ∇λ 1 · r 1 = 1 (12) Using the equation from the previous cell, we finally have d dξ  ρ u  = r 1 =  − ρ c 1  (13) 1.1.1 Expressing u in terms of ξ The simpler of these equations is du dξ = 1 (14) which may be solved to give u in terms of ξ u = ul + (ξ − ξ1) = ξ + c (15) 2 1.1.2 Expressing ρ in terms of ξ For the second state variable ρ, we appeal to the constancy of the first Riemann invariant w 1 = ul + c ln ρl accross the wave. from this we have ρ = exp  w 1 − u c  = ρl exp  ξ − ξ1 c  (16) 1.1.3 Simple rarefaction Riemann problem For illustration, we can construct the simple rarefaction for the test problem ρl = 2 ul = −1 c = 0.7 ur = 1 (17) We have w 1 = −0.5148, ρr = 0.1149, and hence the simple wave is paramterised along the integral curve between ξ1 = −1.7 and ξ2 = 0.3 going from left to right. We may construct the solution using ξ = x/t from u = ξ + c ρ = ρl exp  ul − u c  (18) In the example below we show the solution to the above problem at t