Perfectly Matched Layers
LINEAR CASE
Dispersive waves : Riemann Invariants are used for boundary
conditions
![]() |
xℓ = 0, xr = 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
![](./perfectly-matched-layers_files/GIF1.gif)
LINEAR CASE
Dispersive waves: PML are used
![]() |
xℓ = 0, xr = 20 δx = 0.02
CF L = 0.1
Relaxation parameter λ = 1000
dispersive parameter μ = 0.25, Froude number F = 0.3
Domain:
xℓ = 0, xr = 20, δx = 0.02, CFL = 0.1
PML parameters
![L 4
σ (x) = min (0, (x - 5)) ,
σR (x) = max (0, (x - 15))4.](./perfectly-matched-layers_files/GIF21x.png)
Red dash-dot vertical lines show the PML regions
σL (x), σR(x) are plotted with green lines
Click on the image to restart the test animation
![](./perfectly-matched-layers_files/GIF2.gif)
LINEAR CASE
Incoming waves: PML are used
Income plane wave profile:
![Vex = R (p )cos (kx - ωt ) - IM (p) sin(kx - ωt).](./perfectly-matched-layers_files/GIF30x.png)
![R](./perfectly-matched-layers_files/cmsy10-e-52.png)
![I](./perfectly-matched-layers_files/cmsy10-e-49.png)
![M](./perfectly-matched-layers_files/cmsy10-e-4d.png)
Vex = (h,u,η,w)
cmax – maximal eigenvalue of matrix L
p – eigenvector corresponding to cmax
![( 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](./perfectly-matched-layers_files/GIF31x.png)
Dispersive parameter μ = 0.1
Froude number: F = 0.1
Relaxation parameter: λ = 1000
Domain:
xℓ = 0, xr = 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 )) .](./perfectly-matched-layers_files/GIF32x.png)
Red dash-dot vertical lines show the PML regions
σL (x), σR(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
![](./perfectly-matched-layers_files/GIF3.gif)
NONLINEAR CASE
Outgoing solitary wave
![]() |
Dispersive parameter μ = 0.7
Nonlinearity parameter ε = 0.3
Froude number: F = 0.5
Relaxation parameter: λ = 1000
Domain:
xℓ = 0, xr = 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.](./perfectly-matched-layers_files/GIF41x.png)
Red dash-dot vertical lines show the PML regions
σL (x), σR(x) are plotted with green lines
Click on the image to restart the test animation
![](./perfectly-matched-layers_files/GIF4.gif)
NONLINEAR CASE
Incoming solitary wave
![]() |
Dispersive parameter μ = 0.7
Nonlinearity parameter ε = 0.2
Froude number: F = 0.5
Relaxation parameter: λ = 1000
Domain:
xℓ = 0, xr = 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.](./perfectly-matched-layers_files/GIF51x.png)
Red dash-dot vertical lines show the PML regions
σL (x), σR(x) are plotted with green lines
Click on the image to restart the test animation
![](./perfectly-matched-layers_files/GIF5.gif)