Notes: Outline. Diffusive flux. Notes: Notes: Advection-diffusion

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1 Outline This lecture Diffusion and advection-diffusion Riemann problem for advection Diagonalization of hyperbolic system, reduction to advection equations Characteristics and Riemann problem for acoustics Reading: Chapter 3 Recall: Some slides have section numbers on footer. $CLAW/book Examples from the book. Gallery of applications. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 Diffusive flux q(x, t) concentration β diffusion coefficient (β > 0) diffusive flux βq x (x, t) q t + f x 0 diffusion equation: q t (βq x ) x βq xx (if β const). Heat equation: Same form, where q(x, t) density of thermal energy κt (x, t), T (x, t) temperature, κ heat capacity, flux βt (x, t) (β/κ)q(x, t) q t (x, t) (β/κ)q xx (x, t). R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.2 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.2 Advection-diffusion q(x, t) concentration that advects with velocity u and diffuses with coefficient β: Advection-diffusion equation: flux uq βq x. q t + uq x βq xx. If β > 0 then this is a parabolic equation. Advection dominated if u/β (the Péclet number) is large. Fluid dynamics: parabolic terms arise from thermal diffusion and diffusion of momentum, where the diffusion parameter is the viscosity. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.2 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.2

2 The Riemann problem The Riemann problem consists of the hyperbolic equation under study together with initial data of the form { ql if x < 0 q(x, 0) q r if x 0 Piecewise constant with a single jump discontinuity from q l to q r. The Riemann problem is fundamental to understanding The mathematical theory of hyperbolic problems, Godunov-type finite volume methods Why? Even for nonlinear systems of conservation laws, the Riemann problem can often be solved for general q l and q r, and consists of a set of waves propagating at constant speeds. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8 The Riemann problem for advection The Riemann problem for the advection equation q t + uq x 0 with { ql if x < 0 q(x, 0) q r if x 0 has solution { ql if x < ut q(x, t) q(x ut, 0) if x ut consisting of a single wave of strength W q r q l propagating with speed s u. q r R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8 Riemann solution for advection q(x, T ) x t plane q(x, 0) R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8

3 Discontinuous solutions Note: The Riemann solution is not a classical solution of the PDE q t + uq x 0, since q t and q x blow up at the discontinuity. Integral form: d x2 q(x, t) dx uq(x, t) uq(x 2, t) dt x Integrate in time from t to t 2 to obtain x2 x q(x, t 2 ) dx t2 t x2 x uq(x, t) dt q(x, t ) dx t2 t uq(x 2, t) dt. The Riemann solution satisfies the given initial conditions and this integral form for all x 2 > x and t 2 > t 0. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.7 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.7 Discontinuous solutions Vanishing Viscosity solution: The Riemann solution q(x, t) is the limit as ɛ 0 of the solution q ɛ (x, t) of the parabolic advection-diffusion equation q t + uq x ɛq xx. For any ɛ > 0 this has a classical smooth solution: R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec..6 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec..6 Diagonalization of linear system Consider constant coefficient linear system q t + Aq x 0. Suppose hyperbolic: Real eigenvalues λ λ 2 λ m, Linearly independent eigenvalues r, r 2,..., r m. Let R r r 2 r m m m matrix of eigenvectors. Then Ar p λ p r p means that AR RΛ where λ λ 2 Λ... diag(λ, λ 2,..., λ m ). λ m AR RΛ A RΛR and R AR Λ. Similarity transformation with R diagonalizes A. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.9 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.9

4 Diagonalization of linear system Consider constant coefficient linear system q t + Aq x 0. Multiply system by R : R q t (x, t) + R Aq x (x, t) 0. Introduce RR I: R q t (x, t) + R ARR q x (x, t) 0. Use R AR Λ and define w(x, t) R q(x, t): w t (x, t) + Λw x (x, t) 0. Since R is constant! This decouples to m independent scalar advection equations: w p t (x, t) + λp w p x(x, t) 0. p, 2,..., m. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.9, 3. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.9, 3. Solution to Cauchy problem Suppose q(x, 0) q(x) for < x <. From this initial data we can compute data w (x) R q(x) The solution to the decoupled equation w p t + λp wx p 0 is w p (x, t) w p (x λ p t, 0) w p (x λ p t). Putting these together in vector gives w(x, t) and finally q(x, t) Rw(x, t). We can rewrite this as m m q(x, t) w p (x, t) r p w p (x λ p t) r p p p R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3. Linear acoustics Example: Linear acoustics in a d gas tube p p(x, t) pressure perturbation q u u(x, t) velocity Equations: p t + κu x 0 Change in pressure due to compression ρu t + p x 0 Newton s second law, F ma where K bulk modulus, and ρ unperturbed density of gas. Hyperbolic system: p 0 κ p + 0. u /ρ 0 u t x R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.9.

5 Eigenvectors for acoustics u0 K A 0 /ρ 0 u 0 (acoustics relative to flow with speed u 0 ) Eigenvectors: r ρ0 c 0, r 2 ρ0 c 0. Check that Ar p λ p r p, where λ u 0 c 0, λ 2 u 0 + c 0. with c 0 K 0 /ρ 0 K 0 ρ 0 c 2 0. Note: Eigenvectors are independent of u 0. Let Z 0 ρ 0 c 0 K 0 ρ 0 impedance. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.8 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 2.8 Physical meaning of eigenvectors Eigenvectors for acoustics: r ρ0 c 0 Z0, r 2 ρ0 c 0 Z0. Consider a pure -wave (simple wave), at speed λ c 0, If q (x) q + w (x)r then q(x, t) q + w (x λ t)r Variation of q, as measured by q x or q q(x + x) q(x) is proportional to eigenvector r, e.g. q x (x, t) w x(x λ t)r R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.4, 3.5 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.4, 3.5 Physical meaning of eigenvectors Eigenvectors for acoustics: r ρ0 c 0 Z0, r 2 ρ0 c 0 Z0. In a simple -wave (propagating at speed λ c 0 ), px Z0 β(x) u x The pressure variation is Z 0 times the velocity variation. Similarly, in a simple 2-wave (λ 2 c 0 ), px Z0 β(x) u x The pressure variation is Z 0 times the velocity variation. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5

6 Acoustic waves q(x, 0) p (x) 0 p (x) 2Z 0 Z0 + p (x) Z0 2Z 0 w (x, 0)r + w 2 (x, 0)r 2 p (x)/2 p (x)/(2z 0 ) + p (x)/2 p (x)/(2z0 ). R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5 Solution by tracing back on characteristics The general solution for acoustics: q(x, t) w (x λ t, 0)r + w 2 (x λ 2 t, 0)r 2 w (x + c 0 t, 0)r + w 2 (x c 0 t, 0)r 2 Recall that w(x, 0) R q(x, 0), i.e. w (x, 0) l q(x, 0), w 2 (x, 0) l 2 q(x, 0) where l and l 2 are rows of R. R l l 2 Note: l and l 2 are left-eigenvectors of A: l p A λ p l p since R A ΛR. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5 Solution by tracing back on characteristics The general solution for acoustics: q(x, t) w (x λ t, 0)r + w 2 (x λ 2 t, 0)r 2 w (x + c 0 t, 0)r + w 2 (x c 0 t, 0)r 2 (x, t) x λ 2 t x c 0 t x λ t x + c 0 t R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.6 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.6

7 Solution by tracing back on characteristics The general solution for acoustics: q(x, t) w (x λ t, 0)r + w 2 (x λ 2 t, 0)r 2 q(x, t) w 2 constant w constant w 2 (x λ 2 t, 0) l 2 q(x λ 2 t, 0) w (x λ t, 0) l q(x λ t, 0) R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.5 Riemann Problem Special initial data: q(x, 0) { ql if x < 0 q r if x > 0 Example: Acoustics with bursting diaphram (u l u r 0) Pressure: Acoustic waves propagate with speeds ±c. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec Riemann Problem for acoustics Waves propagating in x t space: Left-going wave W q m q l and right-going wave W 2 q r q m are eigenvectors of A. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.8

8 Riemann Problem for acoustics In x t plane: q(x, t) w (x + ct, 0)r + w 2 (x ct, 0)r 2 Decompose q l and q r into eigenvectors: q l wl r + wl 2 r2 q r wrr + wrr 2 2 Then q m w rr + w 2 l r2 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.9 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.9 Riemann Problem for acoustics q l wl r + wl 2 r2 q r wrr + wrr 2 2 Then q m wrr + wl 2 r2 So the waves W and W 2 are eigenvectors of A: W q m q l (wr wl )r W 2 q r q m (wr 2 wl 2 )r2. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.9 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.9 Riemann solution for a linear system Linear hyperbolic system: q t + Aq x 0 with A RΛR. General Riemann problem data q l, q r lr m. Decompose jump in q into eigenvectors: q r q l m α p r p p Note: the vector α of eigen-coefficients is α R (q r q l ) R q r R q l w r w l. Riemann solution consists of m waves W p lr m : W p α p r p, propagating with speed s p λ p. R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.9 R.J. LeVeque, University of Washington IPDE 20, June 22, 20 FVMHP Sec. 3.9