Perfectly Matched Layers

LINEAR CASE

Dispersive waves : Riemann Invariants are used for boundary conditions

2 h (0,x ) = exp { - (x - 6) }, u(0, x) = 0, η(0, x) = h0 (x), w (0, x) = 0.

x_ℓ = 0, x_r = 12, δx = 0.02
dispersive parameter μ = 0.25, Froude number F = 0.5
CF L = 0.5
Dispersion is added: λ = 1000; Evolution of η, h coincide

Click on the image to restart the test animation

LINEAR CASE
Dispersive waves: PML are used

h0 (x ) = exp { - (x - 6)2}, u0(x ) = 0, η0 (x) = h0 (x), w0 (x ) = 0.

x_ℓ = 0, x_r = 20 δx = 0.02

CF L = 0.1

Relaxation parameter λ = 1000
dispersive parameter μ = 0.25, Froude number F = 0.3
Domain:
x_ℓ = 0, x_r = 20, δx = 0.02, CFL = 0.1

PML parameters

L 4 σ (x) = min (0, (x - 5)) , σR (x) = max (0, (x - 15))4.

Red dash-dot vertical lines show the PML regions

σ^L (x), σ^R(x) are plotted with green lines

Click on the image to restart the test animation

LINEAR CASE
Incoming waves: PML are used

Income plane wave profile:

Vex = R (p )cos (kx - ωt ) - IM (p) sin(kx - ωt).

denote real and imaginary parts; ω∕k = c_max

V_ex = (h,u,η,w)

c_max – maximal eigenvalue of matrix L

p – eigenvector corresponding to c_max

( ω ) 1 - -- 1 0 0 | k | || -1-- λ- ω- λ- || | 2 + 1 - - 0 | L = || F 3 k 3 || . | 0 0 1 - ω- -i-- | | k μk | |( iλ iλ ω |) ---- 0 - ---- 1 - -- μk μk k

Dispersive parameter μ = 0.1
Froude number: F = 0.1
Relaxation parameter: λ = 1000
Domain:
x_ℓ = 0, x_r = 20, δx = 0.02, CFL = 0.1

PML parameters

σL (x ) = min (0,0.1 (x - 4))2, R 2 σ (x ) = max (0, 0.1(x - 16 )) .

Red dash-dot vertical lines show the PML regions

σ^L (x), σ^R(x) are plotted with green line

χ(x) is shown in blue together with the depth h(t,x)

Click on the image to restart the test animation

NONLINEAR CASE

Outgoing solitary wave

( √ --- ) ( ) 2 3 ε 1 h0(x)=1 + ε sech -------(x - St - 12 ) ,u0(x ) = S 1 - ------- 2μF S h0(x ) 1 √ ------ S = --- 1 + ε F η(0, x) = h (x ), w (0,x ) = - h (x )-∂-u (x ). 0 0 ∂x 0

Dispersive parameter μ = 0.7
Nonlinearity parameter ε = 0.3
Froude number: F = 0.5
Relaxation parameter: λ = 1000
Domain:
x_ℓ = 0, x_r = 30, δx = 0.05, CFL = 0.1

PML parameters

L 2 σ (x ) = min (0,0.1 (x - 6)) , σR (x ) = max (0, 0.1(x - 20 ))2.

Red dash-dot vertical lines show the PML regions

σ^L (x), σ^R(x) are plotted with green lines

Click on the image to restart the test animation

NONLINEAR CASE

Incoming solitary wave

( √ --- ) ( ) 2 3ε 1 h(t,x)=1 + ε sech -------(x - St + 5) ,u (t,x) = S 1 - ------- 2μF S h0 (x) 1 √ ------ S = --- 1 + ε F

Dispersive parameter μ = 0.7
Nonlinearity parameter ε = 0.2
Froude number: F = 0.5
Relaxation parameter: λ = 1000
Domain:
x_ℓ = 0, x_r = 30, δx = 0.05, CFL = 0.1
PML parameters

L 2 σ (x ) = min (0,0.1 (x - 6)) , σR (x ) = max (0, 0.1(x - 20 ))2.

Red dash-dot vertical lines show the PML regions

σ^L (x), σ^R(x) are plotted with green lines

Click on the image to restart the test animation