29-Jul-2021 14:55:50
schroedinger_linear_pde_test
MATLAB/Octave version 9.9.0.1467703 (R2020b)
Solve the linear Schroedinger PDE
using the method of lines
and various ODE solvers.
Parameter values:
alpha = 0.5
c1 = 1
c2 = 0.1
delta = 25
gamma = 1
t0 = 0
xmin = -20
xmax = 80
schroedinger_linear_ode45
MATLAB/Octave version 9.9.0.1467703 (R2020b)
Solve the linear Schroedinger PDE using ode45()
Solve the linear Schroedinger equation in 1D,
du/dt = i uxx + i gamma |u|^2 u
over the spatial interval:
-20 <= x <= 80
and time interval
0 <= t <= 44.0
with Neumann boundary conditions:
u'(-20) = 0.0
u'(80) = 0.0
and parameter:
gamma = 1
and initial condition
u(0,x) = sqrt(2 alpha/gamma) *
( exp(0.5 i c1 x ) sech( sqrt(alpha) x )
+ exp(0.5 i c2 (x-delta) sech( sqrt(alpha)(x-delta) ).
and NX equally spaced nodes in X,
using ode45().
Use 201 nodes, with dx=0.5 in space interval [-20,80]
Time interval is [0,10]
Number of (variable size) time steps = 273
Graphics saved as "schroedinger_linear_ode45_initial.png"
Graphics saved as "schroedinger_linear_ode45_midway.png"
Graphics saved as "schroedinger_linear_ode45_final.png"
Graphics saved as "schroedinger_linear_ode45_conservation.png"
schroedinger_linear_ode45
Normal end of execution.
schroedinger_linear_midpoint
MATLAB/Octave version 9.9.0.1467703 (R2020b)
Solve the linear Schroedinger PDE using midpoint()
Solve the linear Schroedinger equation in 1D,
du/dt = i uxx
over the spatial interval:
-20 <= x <= 80
using 201 equally spaced spatial nodes
and time interval
0 <= t <= 44.0
using 250 equally spaced time steps
with Neumann boundary conditions:
u'(-20) = 0.0
u'(80) = 0.0
and parameter:
gamma = 1
and initial condition
u(0,x) = sqrt(2 alpha/gamma) *
( exp(0.5 i c1 x ) sech( sqrt(alpha) x )
+ exp(0.5 i c2 (x-delta) sech( sqrt(alpha)(x-delta) ).
using midpoint().
Use 201 nodes, with dx=0.5 in space interval [-20,80]
Use 250 steps, with dt=0.04 in time interval [0,10]
Graphics saved as "schroedinger_linear_midpoint_initial.png"
Graphics saved as "schroedinger_linear_midpoint_midway.png"
Graphics saved as "schroedinger_linear_midpoint_final.png"
Graphics saved as "schroedinger_linear_midpoint_conservation.png"
schroedinger_linear_midpoint
Normal end of execution.
schroedinger_linear_pde_test
Normal end of execution.
29-Jul-2021 14:57:31