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

2024
Views:

480

Pages:

21
Author:

Cali Bowman

Scalar Riemann Problem qt + ūqx = 0 subject to initial conditions q qr q(x, 0) = x<0 x>0 x/t = ū t Solution to the Riemann problem q Solution : qr Scalar constant coefficient advection Discontinuity propagates at speed t=1 t=0 and has strength Solving constant coefficient linear systems qt + Aqx = 0, A 2 Rm⇥m We assume that A has a complete set of eigenvectors and real eigenvalues and so can be written as A = R⇤R R = [r , r , . . . r ] 1 2 m 1 ⇤ = diag( 1 , 2 ,... Examples : Linearized shallow water wave equations, constant coefficient acoustics, ... m ) Solving a constant coefficient system ! q t + A qx = 0 qt + R⇤R 1 Define characteristic variables (x, t) = R 1 q(x, t), A 2 Rm⇥m qx = 0, R m Assume that A is diagonalizable as (x, 0) = R 1 q(x, 0) Characteristic equations decouple into m scalar equations : p ⇥t + p ⇥xp = 0, p = 1, 2, . . . , m Solution to characteristic equations are given by ⇥ (x, t) = ⇥ (x p p p t, 0) Solving a constant coefficient systems p ⇥t q t + A qx = 0 + p p ⇥x =0 Solution for general initial conditions q(x, 0) : m q(x, t) = R ⇥(x, t) = ⇥ (x, t) r p p ⇥ (x p p=1 m = q(x, t) p t, 0) r p p=1 t `p q(x x ⇥ 3 (x 3 t, 0) ⇥ 2 (x 2 t, 0) ⇥ 1 (x 1 t, 0) p t, 0) = ! p (x p t, 0) Riemann problem for systems Assume a constant coefficient system : with piecewise constant initial data : q(x, 0) = q qr x<0 x>0 which can be decomposed as : q` = 3 X p=1 p p !` r qr = 3 X p=1 p p !r r Riemann problem for systems t>0 Piecewise constant initial data t=0 x1 q q(x1 , t) = qr x=0 r + 1 1 r + 2 2 m r p=1 r m p p q = 3 3 qr = p p r r p=1 = q` Riemann problem for systems q t + A qx = 0 t>0 t=0 x2 q q(x2 , t) = 1 1 r r qr + r + 2 2 m r p=1 r m p p q = 3 3 qr = p p r r p=1 Riemann problem for systems q t + A qx = 0 t>0 t=0 x3 q q(x3 , t) = 1 1 r r + 2 2 r r m r p=1 + 3 3 r m p p q = qr qr = p p r r p=1 Riemann problem for systems q t + A qx = 0 t>0 t=0 q x4 qr q(x4 , t) = 1 1 r r + 2 2 r r m m p p q = r p=1 + 3 3 r r qr = p p r r p=1 = qr Riemann problem for systems q t + A qx = 0 t>0 t=0 q qr q(x1 , t) = 1 1 r + 2 2 r + 3 3 q(x2 , t) = 1 1 r r + 2 2 r + 3 3 q(x3 , t) = 1 1 r r + 2 2 r r + 3 3 q(x4 , t) = 1 1 r r + 2 2 r r + 3 3 r r r r r Riemann problem for systems q t + A qx = 0 t>0 t=0 q (x, t) p q qr } ??? Riemann problem for systems q t + A qx = 0 } q (x, t) p q(x1 , t) = 1 1 q(x2 , t) = 1 1 r r q(x2 , t) r + 2 2 + 2 2 q(x1 , t) = ( r + 3 3 r + 3 3 1 r r r 1 )r 1 Riemann problem for systems q t + A qx = 0 } q (x, t) p q(x2 , t) = 1 1 r r + 2 2 q(x3 , t) = 1 1 r r + 2 2 r r q(x3 , t) q(x2 , t) = ( r + 3 3 + 3 3 2 r r r 2 )r 2 Riemann problem for systems q t + A qx = 0 q (x, t) p q(x3 , t) = 1 1 r r q(x4 , t) = 1 1 r r q(x4 , t) + 2 2 r r + 3 3 + 2 2 r r + 3 3 r r q(x3 , t) = ( 3 r r 3 )r 3 } Riemann problem for systems q t + A qx = 0 q p (x, t) 3 p=1 3 ⇥ R = qr p p r p=1 q p ⇥ ) rp { q(x1 , t) = { q(x4 , t) (⇥rp Riemann problem for systems Solving the Riemann problem for linear problem q t + A qx = 0 (1) Compute eigenvalues and eigenvectors of matrix A (2) Compute characteristic variables by solving R = qr q (3) Use eigenvalues or “speeds” to determine piecewise constant solution q(x, t) = q⇧ + p p r p: = p x/t Numerical solution qt + f (q)x = 0, Qn+1 i = Qi f (q) = Aq t n Fi+1/2 x Qni Fin 1/2 ⇥ 2 2 wave ↵ r travels at speed 2 > 0 q⇤ 1 Qi xi 1/2 Decompose jump in Q at the interface into waves : q ⇤ = Qi Fi 1/2 1 1 3 3 2 2 + ↵ r = Q ↵ r ↵ r 1 i Z t+ t 1 ⇡ f (q(xi 1/2 , t))dt = Aq ⇤ t t Example : Linearized shallow water qt + A qx = 0, A= ✓ U g H U ◆ , q= ✓ h u ◆ Characteristic information : Eigenvalues : Eigenvectors : 1 =U r1 = ✓ p gH, ◆ p gH , g 2 p = U + gH ✓ p ◆ gH 2 r = g Example : Linearized shallow water Characteristic variables : 1 Define : ✓ R = qr = qr ↵1 ↵2 ◆ q` 1 = 2gH ✓ ◆ ✓ 8 h` > > q` = > > u` > > < 1 1 q + ↵ r ` q(x, t) = > ◆ ✓ > > > hr > > : qr = u ` ! p gH p gH 2 q = = H H hr ur ◆✓ p 1 2 h` u` ◆ gH x/t < U p p U gH < x/t < U + gH x/t > U + p gH Linear shallow water wave equations Initial height and velocity W1 W2 !