## Numerical solution of PDE:s, Part 7: 2D Schrödinger equation

Haven’t been posting for a while, but here’s something new… Earlier I showed how to solve the 1D Schrödinger equation numerically in different situations. Now I’m going to show how to calculate the evolution of a 2D wavepacket in a potential energy field that has been constructed to mimic the classical “two-slit experiment” which shows how the mechanics of low-mass particles like electrons can exhibit interference similar to the mechanics of classical waves (sound, light, water surface, and so on).

A 2D Schrödinger equation for a single particle in a time-independent background potential V(x,y) is

Where the particle mass has been set to 1 and the Planck’s constant to $2\pi$.

To solve this numerically, we need the Crank-Nicolson method, as was the case when solving the 1D problem. More specifically, the linear system to be solved is

with

where the wavefunction now has two position indices and one time index, and the potential energy has only two position indices.

To form a model of the two-slit experiment, we choose a domain 0 < x < 6; 0 < y < 6 and make a potential energy function defined by

IF (x < 2.2 OR x > 3.8 OR (x > 2.7 AND x < 3.3)) THEN IF (3.7 < y < 4) THEN V(x,y) = 30

IF (x < 0.5 OR x > 5.5 OR y < 0.5 OR y > 5.5) THEN V(x,y) = 30

Otherwise V(x,y) = 0.

which corresponds to having hard walls surrounding the domain and a barrier with two holes around the line y = 3.85

For an initial condition, we choose a Gaussian wavepacket that has a nonzero expectation value of the momentum in y-direction:

An R-Code that solves this problem for a time interval 0 < t < 1 is

library(graphics) #load the graphics library needed for plotting

lx <- 6.0 #length of the computational domain (both x and y)
lt <- 1.0 #length of the simulation time interval
nx <- 47 #number of discrete lattice points in x and y directions
nt <- 60 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell, same in x and y directions
dt <- lt/nt #length of timestep

V = matrix(nrow=nx,ncol=nx) #potential energies at discrete xy points

for(j in c(1:nx)) { #construct the potential field with a double-slit barrier
for(k in c(1:nx)) {
V[j,k] = 0+0i
if((j*dx<2.2)||(j*dx>3.8)||((j*dx>2.7) && (j*dx<3.3))) {
if((k*dx>3.7) && (k*dx<4.0)) {
V[j,k] = 30+0i #No significant density is going to go through these barriers
}
}
if((j*dx>5.5) || (j*dx<0.5) || (k*dx>5.5) || (k*dx<0.5)) {
V[j,k] = 30+0i
}
}
}

kappa1 = (1i)*dt/(2*dx*dx) #an element needed for the matrices
kappa2 <- c(1:(nx*nx)) #another element

for(j in c(1:nx)) {
for(k in c(1:nx)) {
kappa2[(j-1)*nx+k] <- kappa1*2*dx*dx*V[j,k]
}
}

psi = c(1:(nx*nx)) #array for the wave function values

for(j in c(1:nx)) {
for(k in c(1:nx)) {
psi[(j-1)*nx+k] = exp(-2*(k*dx-1.7)*(k*dx-1.7)-2*(j*dx-3)*(j*dx-3)+(2i)*k*dx) #Gaussian initial wavefunction
if((j*dx > 5.5)||(j*dx < 0.5)||(k*dx > 5.5)||(k*dx < 0.5)) {
psi[(j-1)*nx+k] = as.complex(0)
}
}
}

xaxis <- c(1:nx)*dx #the x values corresponding to the discrete lattice points
yaxis <- c(1:nx)*dx #the y values

A = matrix(nrow=nx*nx,ncol=nx*nx) #matrix for forward time evolution
B = matrix(nrow=nx*nx,ncol=nx*nx) #matrix for backward time evolution
P = matrix(nrow=nx,ncol=nx) #matrix for the solution after time stepping
IP = matrix(nrow=4*nx,ncol=4*nx) #matrix for the higher resolution image obtained by interpolation

for(j in c(1:(nx*nx))) { #Set the values for time evolution matrix elements
for(k in c(1:(nx*nx))) {
A[j,k]=0
B[j,k]=0
if(j==k) {
A[j,k] = 1 + 4*kappa1 + kappa2[j]
B[j,k] = 1 – 4*kappa1 – kappa2[j]
}
if((k==j+1) || (k==j-1)) {
A[j,k] = -kappa1
B[j,k] = kappa1
}
if((k==j+nx)||(k==j-nx)) {
A[j,k] = -kappa1
B[j,k] = kappa1
}
}
}

for(k in c(1:nt)) { #main time stepping loop

sol <- solve(A,B%*%psi) #solve the system of equations

for(l in c(1:(nx*nx))) {
psi[l] <- sol[l]
}

for(l in c(1:nx)) {
for(m in c(1:nx)) {
P[l,m] = abs(psi[(l-1)*nx + m])*abs(psi[(l-1)*nx + m]) #square of the absolute value of wave function
if(abs(V[l,m]) > 5) P[l,m] = 2
}
}
for(l in c(1:(nx-1))) {
for(m in c(1:(nx-1))) { #make a bitmap with 4 times more pixels, using linear interpolation
IP[4*l-3,4*m-3] = P[l,m]
IP[4*l-2,4*m-3] = P[l,m]+0.25*(P[l+1,m]-P[l,m])
IP[4*l-1,4*m-3] = P[l,m]+0.5*(P[l+1,m]-P[l,m])
IP[4*l,4*m-3] = P[l,m]+0.75*(P[l+1,m]-P[l,m])
}
}

for(l in c(1:(4*nx))) {
for(m in c(1:(nx-1))) {
IP[l,4*m-2] = IP[l,4*m-3]+0.25*(IP[l,4*m+1]-IP[l,4*m-3])
IP[l,4*m-1] = IP[l,4*m-3]+0.5*(IP[l,4*m+1]-IP[l,4*m-3])
IP[l,4*m] = IP[l,4*m-3]+0.75*(IP[l,4*m+1]-IP[l,4*m-3])
}
}

jpeg(file = paste(“plot_abs_”,k,”.jpg”,sep=””)) #save the image
image(IP, zlim = c(0,0.15))
dev.off()

}

The code produces a sequence of image files, where the probability density is plotted with colors, as an output. Some representative images from this sequence (converted to grayscale) is shown below:

A video of the time evolution is shown below:

The treshold for maximum white color has been chosen to be quite low, to make the small amount of probability density that crosses the barrier visible.

The discrete grid of points has been made quite coarse here to keep the computation time reasonable, and the resolution has been increased artificially by using linear interpolation between the discrete points.

So, now we’ve seen how to solve the motion of 2D wavepackets moving around obstacles. In the next numerical methods post, I’ll go through the numerical solution of a nonlinear PDE.

## Numerical solution of PDE:s, Part 4: Schrödinger equation

In the earlier posts, I showed how to numerically solve a 1D or 2D diffusion or heat conduction problem using either explicit or implicit finite differencing. In the 1D example, the relevant equation for diffusion was

and an important property of the solution was the conservation of mass,

i.e. the integral of the concentration field over whole domain stays constant.

Next, I will show how to integrate the 1D time-dependent Schrödinger equation, which in a nondimensional form where we set $\hbar = 1$ and $m = 1$ reads:

Here $i$ is the imaginary unit and $V(x)$ is the potential energy as a function of $x$. The solutions of this equation must obey a conservation law similar to the mass conservation in the diffusion equation, the conservation of norm:

where the quantity $|\Psi (x,t)|$ is the modulus of the complex-valued function $\Psi (x,t)$ . This property of the solution is also called unitarity of the time evolution.

Apart from the TDSE, another way to represent the time development of this system is to find the normalized solutions $\psi_0 (x)$, $\psi_1 (x)$, $\psi_2 (x) \dots$ of the time-independent Schrödinger equation

and write the initial state $\Psi (x,0)$ as a linear combination of those basis functions:

This is possible because the solutions of the time-independent equation form a basis for the set of acceptable wave functions $\psi (x)$. Then, every term in that eigenfunction expansion is multiplied by a time dependent phase factor $\exp(-iE_n t)$:

The numbers $E_n$ are the eigenvalues corresponding to the solutions $\psi_n (x)$ and the function $\psi_0 (x)$ is called the ground state corresponding to potential $V(x)$, while the functions $\psi_1 (x)$ is the first excited state and so on.

The Schrödinger equation can’t be discretized by using either the explicit or implicit method that we used when solving the diffusion equation. The method is either numerically unstable or doesn’t conserve the normalization of the wave function (or both) if you try to do that. The correct way to discretize the Schrödinger equation is to replace the wave function with a discrete equivalent

and the potential energy function $V(x)$ with $V_{i;j}$ (or $V_i$ in the case of time-independent potential), and write an equation that basically tells that propagating the state $\Psi_{i;j}$ forward by half a time step gives the same result as propagating the state $\Psi_{i;j+1}$ backwards by half a time step:

Here we have

and

This kind of discretization is called the Crank-Nicolson method. As boundary conditions, we usually set that at the boundaries of the computational domain the wavefunction stays at value zero: $\Psi (0,t) = \Psi (L,t) = 0$ for any value of $t$. In the diffusion problem, this kind of a BC corresponded to infinite sinks at the boundaries, that annihilated anything that diffused through them. In the Schrödinger equation problem, which is a complex diffusion equation, the equivalent condition makes the boundaries impenetrable walls that deflect elastically anything that collides with them.

An R-Code that calculates the time evolution of a Gaussian initial wavefunction

in an area of zero potential:

for a domain 0 < x < 6, a lattice spacing $\Delta x = 0.05$, time interval 0 < t < 2 and time step $\Delta t = 0.01$, is given below:

library(graphics) #load the graphics library needed for plotting

lx <- 6.0 #length of the computational domain
lt <- 2.0 #length of the simulation time interval
nx <- 120 #number of discrete lattice points
nt <- 200 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell
dt <- lt/nt #length of timestep

V = c(1:nx) #potential energies at discrete points

for(j in c(1:nx)) {
V[j] = 0 #zero potential
}

kappa1 = (1i)*dt/(2*dx*dx) #an element needed for the matrices
kappa2 <- c(1:nx) #another element

for(j in c(1:nx)) {
kappa2[j] <- as.complex(kappa1*2*dx*dx*V[j])
}

psi = as.complex(c(1:nx)) #array for the wave function values

for(j in c(1:nx)) {
psi[j] = as.complex(exp(-2*(j*dx-3)*(j*dx-3))) #Gaussian initial wavefunction
}

xaxis <- c(1:nx)*dx #the x values corresponding to the discrete lattice points

A = matrix(nrow=nx,ncol=nx) #matrix for forward time evolution
B = matrix(nrow=nx,ncol=nx) #matrix for backward time evolution

for(j in c(1:nx)) {
for(k in c(1:nx)) {
A[j,k]=0
B[j,k]=0
if(j==k) {
A[j,k] = 1 + 2*kappa1 + kappa2[j]
B[j,k] = 1 - 2*kappa1 - kappa2[j]
}
if((j==k+1) || (j==k-1)) {
A[j,k] = -kappa1
B[j,k] = kappa1
}
}
}

for (k in c(1:nt)) { #main time stepping loop

sol <- solve(A,B%*%psi) #solve the system of equations

for (l in c(1:nx)) {
psi[l] <- sol[l]
}

if(k %% 3 == 1) { #make plots of |psi(x)|^2 on every third timestep
jpeg(file = paste("plot_",k,".jpg",sep=""))
plot(xaxis,abs(psi)^2,xlab="position (x)", ylab="Abs(Psi)^2",ylim=c(0,2))
title(paste("|psi(x,t)|^2 at t =",k*dt))
lines(xaxis,abs(psi)^2)
dev.off()
}
}

The output files are plots of the absolute squares of the wavefunction, and a few of them are shown below.

In the next simulation, I set the domain and discrete step sizes the same as above, but the initial state is:

Which is a Gaussian wave packet that has a nonzero momentum in the positive x-direction. This is done by changing the line

for(j in c(1:nx)) {
psi[j] = as.complex(exp(-2*(j*dx-3)*(j*dx-3))) #Gaussian initial wavefunction
}+(1i)*j*dx

into

for(j in c(1:nx)) {
psi[j] = as.complex(exp(-2*(j*dx-3)*(j*dx-3)+(1i)*j*dx)) #Gaussian initial wavefunction
}

The plots of $|\Psi (x,t)|^2$ for several values of $t$ are shown below

and there you can see how the wave packet collides with the right boundary of the domain and bounces back.

In the last simulation, I will set the potential function to be

which is a harmonic oscillator potential, and with the nondimensional mass $m =1$ and Planck constant $\hbar = 1$ the ground state $\psi _0 (x)$ of this system is

If I’d set the initial state to be $\Psi (x,0) = \psi_0 (x)$, or any other solution of the time-independent SE, the modulus of the wavefunction would not change at all. To get something interesting to happen, I instead set an initial state that is a displaced version of the ground state:

The solution can be obtained with the code shown below:

library(graphics) #load the graphics library needed for plotting

lx <- 6.0 #length of the computational domain
lt <- 3.0 #length of the simulation time interval
nx <- 360 #number of discrete lattice points
nt <- 300 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell
dt <- lt/nt #length of timestep

V = c(1:nx) #potential energies at discrete points

for(j in c(1:nx)) {
V[j] = as.complex(2*(j*dx-3)*(j*dx-3)) #Harmonic oscillator potential with k=4
}

kappa1 = (1i)*dt/(2*dx*dx) #an element needed for the matrices
kappa2 <- c(1:nx) #another element

for(j in c(1:nx)) {
kappa2[j] <- as.complex(kappa1*2*dx*dx*V[j])
}

psi = as.complex(c(1:nx)) #array for the wave function values

for(j in c(1:nx)) {
psi[j] = as.complex(exp(-(j*dx-2)*(j*dx-2))) #Gaussian initial wavefunction, displaced from equilibrium
}

xaxis <- c(1:nx)*dx #the x values corresponding to the discrete lattice points

A = matrix(nrow=nx,ncol=nx) #matrix for forward time evolution
B = matrix(nrow=nx,ncol=nx) #matrix for backward time evolution

for(j in c(1:nx)) {
for(k in c(1:nx)) {
A[j,k]=0
B[j,k]=0
if(j==k) {
A[j,k] = 1 + 2*kappa1 + kappa2[j]
B[j,k] = 1 - 2*kappa1 - kappa2[j]
}
if((j==k+1) || (j==k-1)) {
A[j,k] = -kappa1
B[j,k] = kappa1
}
}
}

for (k in c(1:nt)) { #main time stepping loop

sol <- solve(A,B%*%psi) #solve the system of equations

for (l in c(1:nx)) {
psi[l] <- sol[l]
}

if(k %% 3 == 1) { #make plots of Abs(psi(x))^2 on every third timestep
jpeg(file = paste("plot_",k,".jpg",sep=""))
plot(xaxis,abs(psi)^2, xlab="position (x)", ylab="Abs(Psi)^2",ylim=c(0,2))
title(paste("|psi(x,t)|^2 at t =",k*dt))
lines(xaxis,abs(psi)^2)
lines(xaxis,V)
dev.off()
}
}

and the solution at different values of $t$ look like this (images and video):

Here the shape of the Hookean potential energy is plotted in the same images. So, here you see how the center of the Gaussian wavefunction oscillates around the point $x = 3$, just like a classical mechanical harmonic oscillator does when set free from a position that is displaced from equilibrium.

By changing the code that produces the output images, we can also get a sequence of plots of the imaginary part of the wavefunction:

if(k %% 3 == 1) { #make plots of Im(psi(x)) on every third timestep
jpeg(file = paste("plot_",k,".jpg",sep=""))
plot(xaxis,Im(psi), xlab="position (x)", ylab="Im(Psi)",ylim=c(-1.5,1.5))
title(paste("Im(psi(x,t)) at t =",k*dt))
lines(xaxis,Im(psi))
lines(xaxis,V)
dev.off()
}

and the resulting plots look like this:

## Numerical solution of PDE:s, Part 3: 2D diffusion problem

In the earlier posts related to PDE numerical integration, I showed how to discretize 1-dimensional diffusion or heat conduction equations either by explicit or implicit methods. The 1d model can work well in some situations where the symmetry of a physical system makes the concentration or temperature field practically depends on only one cartesian coordinate and is independent of the position along two other orthogonal coordinate axes.

The 2-dimensional version of the diffusion/heat equation is

assuming that the diffusion is isotropic, i.e. its rate does not depend on direction.

In a discretized description, the function C(x,y,t) would be replaced by a three-index object

assuming that one of the corners of the domain is at the origin. Practically the same can also be done with two indices, as in

where $N_x$ is the number of discrete points in x-direction.

Using the three-index version of the 2D diffusion equation and doing an explicit discretization, we get this kind of a discrete difference equation:

which can be simplified a bit if we have $\Delta x = \Delta y$ .

Below, I have written a sample R-Code program that calculates the evolution of a Gaussian concentration distribution by diffusion, in a domain where the x-interval is 0 < x < 6 and y-interval is 0 < y < 6. The boundary condition is that nothing diffuses through the boundaries of the domain, i.e. the two-dimensional integral of C(x,y,t) over the area $[0,6]\times [0,6]$ does not depend on t.

library(graphics) #load the graphics library needed for plotting

lx <- 6.0 #length of the computational domain in x-direction
ly <- 6.0 #length of the computational domain in y-direction
lt <- 6 #length of the simulation time interval
nx <- 30 #number of discrete lattice points in x-direction
ny <- 30 #number of discrete lattice points in y-direction
nt <- 600 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell in x-direction
dy <- ly/ny #length of one discrete lattice cell in y-direction
dt <- lt/nt #length of timestep

D <- 1.0 #diffusion constant (assumed isotropic)

Conc2d = matrix(nrow=ny,ncol=nx)
DConc2d <- matrix(nrow=ny, ncol=nx) #a vector for the changes in concentration during a timestep
xaxis <- c(0:(nx-1))*dx #the x values corresponding to the discrete lattice points
yaxis <- c(0:(ny-1))*dy #the y values corresponding to the discrete lattice points

kappax <- D*dt/(dx*dx) #a parameter needed in the discretization
kappay <- D*dt/(dy*dy) #a parameter needed in the discretization

for (i in c(1:ny)) {
for (j in c(1:nx)) {
Conc2d[i,j] = exp(-(i*dy-3)*(i*dy-3)-(j*dx-3)*(j*dx-3)) #2D Gaussian initial concentration distribution
DConc2d[i,j] <- 0 #all initial values in DConc vector zeroed
}
}

for (j in c(1:nt)) { #main time stepping loop

for(k in c(1:nx))
{
Conc2d[1,k] <- Conc2d[k,2] #fluxes through the boundaries of the domain are forced to stay zero
Conc2d[nx,k] <- Conc2d[nx-1,k]
}

for(k in c(1:ny)) {
Conc2d[k,1] <- Conc2d[k,2]
Conc2d[k,nx] <- Conc2d[k,nx-1]
}
for (k in c(2:(ny-1))) {
for (l in c(2:(nx-1))) {
DConc2d[k,l] <- kappax*(Conc2d[k,l-1]-2*Conc2d[k,l]+Conc2d[k,l+1]) + kappay*(Conc2d[k-1,l]-2*Conc2d[k,l]+Conc2d[k+1,l]) #time stepping
}
}

for (k in c(2:(ny-1))) {
for (l in c(2:(nx-1))) {
Conc2d[k,l] <- Conc2d[k,l]+DConc2d[k,l] #add the changes to the vector Conc
}
}
k <- 0
l <- 0

if(j %% 3 == 1) { #make plots of C(x,y) on every third timestep
jpeg(file = paste("plot_",j,".jpg",sep=""))
persp(yaxis,xaxis,Conc2d,zlim=c(0,1))
title(paste("C(x,y) at t =",j*dt))
dev.off()
}
}

To plot the distribution C(x,y) with a color map instead of a 3D surface, you can change the line

persp(yaxis,xaxis,Conc2d,zlim=c(0,1))

to

image(yaxis,xaxis,Conc2d,zlim=c(0,0.3))

The two kinds of graphs are shown below for three different values of t.

Figure 1. Surface plots of the time development of a Gaussian mass or temperature distribution spreading by diffusion.

Figure 2. Time development of a Gaussian mass or temperature distribution spreading by diffusion, plotted with a red-orange-yellow color map.

To obtain an implicit differencing scheme, we need to write the discretized equation as a matrix-vector equation where $C_{i;j;k}$ is obtained from $C_{i;j;k+1}$ by backward time stepping. In the square matrix, there is one row (column) for every lattice point of the discrete coordinate system. Therefore, if there are N points in x-direction and N points in y-direction, then the matrix is an $N^2 \times N^2$ – matrix. From this it’s quite obvious that the computation time increases very quickly when the spatial resolution is increased.

Initially, one may think that the equation corresponding to diffusion in a really coarse $3 \times 3$ grid is the following one:

Where I’ve used the denotation $k_x = \frac{D\Delta t}{(\Delta x)^2}$ and $k_y = \frac{D\Delta t}{(\Delta x)^2}$ The problem with this is that there are unnecessary terms $-k_x$ in here, creating a cyclic boundary condition (mass that is diffusing through the boundary described by line x = L reappears from the boundary on the other side, x = 0. The correct algorithm for assigning the nonzero elements $A_{ij}$ of the matrix A is

1. $A_{ij} = 1 + 2k_x + 2k_y$ , when $i = j$
2. $A_{ij} = -k_x$ , when $j=i-1$ AND $i\neq 1$ (modulo $N_x$)
3. $A_{ij} = -k_x$ , when $j=i+1$ AND $i\neq 0$ (modulo $N_x$)
4. $A_{ij} = -k_y$ , when $j=i+N_x$ OR $j=i-N_x$

when you want to get a boundary condition which ensures that anything that diffuses through the boundaries is lost forever. Then the matrix-vector equation in the case of $3 \times 3$ lattice is

Unlike the linear system in the 1D diffusion time stepping, this is not a tridiagonal problem and consequently is slower to solve. An R-Code that produces a series of images of the diffusion process for a Gaussian concentration distribution in a $15 \times 15$ discrete lattice is given below.

library(graphics) #load the graphics library needed for plotting

lx <- 6.0 #length of the computational domain in x-direction
ly <- 6.0 #length of the computational domain in y-direction
lt <- 6 #length of the simulation time interval
nx <- 15 #number of discrete lattice points in x-direction
ny <- 15 #number of discrete lattice points in y-direction
nt <- 180 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell in x-direction
dy <- ly/ny #length of one discrete lattice cell in y-direction
dt <- lt/nt #length of timestep

D <- 1.0 #diffusion constant

C = c(1:(nx*ny))
Cu = c(1:(nx*ny))
Conc2d = matrix(nrow=ny,ncol=nx)
xaxis <- c(0:(nx-1))*dx #the x values corresponding to the discrete lattice points
yaxis <- c(0:(ny-1))*dy #the y values corresponding to the discrete lattice points

kappax <- D*dt/(dx*dx) #a parameter needed in the discretization
kappay <- D*dt/(dy*dy) #a parameter needed in the discretization

A = matrix(nrow=(nx*ny),ncol=(nx*ny))

for(i in c(1:(nx*ny))) {
for(j in c(1:(nx*ny))) {
A[i,j] <- 0
if(i==j) A[i,j] <- 1+2*kappax+2*kappay
if(j==i+1 && (i%%nx != 0)) A[i,j] <- -kappax
if(j==i-1 && (i%%nx != 1)) A[i,j] <- -kappax
if(j==i+nx) A[i,j] <- -kappay
if(j==i-nx) A[i,j] <- -kappay
}
}

for (i in c(1:ny)) {
for (j in c(1:nx)) {
Conc2d[i,j] <- exp(-2*(i*dy-3)*(i*dy-3)-2*(j*dx-3)*(j*dx-3)) #2D Gaussian initial concentration distribution
}
}

for (j in c(1:nt)) { #main time stepping loop

for(k in c(1:ny)) {
for(l in c(1:nx)) {
C[(k-1)*nx+l] <- Conc2d[k,l]
}
}

Cu <- solve(A,C)

for(k in c(1:ny)) {
for(l in c(1:nx)) {
Conc2d[k,l] <- Cu[(k-1)*nx+l]
}
}

k <- 0
l <- 0

if(j %% 3 == 1) { #make plots of C(x,y) on every third timestep
jpeg(file = paste("plot_",j,".jpg",sep=""))
persp(y=xaxis,x=yaxis,z=Conc2d,zlim=c(0,1))
title(paste("C(x) at t =",j*dt))
dev.off()
}
}

Note that the function C(x,y) approaches a constant zero function as t increases, try modifying the code yourself to make a boundary condition that doesn’t let anything diffuse through the boundaries!

## Numerical solution of PDE:s, Part 2: Implicit method

In the previous blog post, I showed how to solve the diffusion equation

using the explicit method, where the equation is converted to a discrete one

which can be simplified by using the notation

The problem with this approach is that if we want to have a high resolution, i.e. $\Delta x$ is very small, the timestep $\Delta t$ has to be made even much smaller to keep the procedure numerically stable. The stability of this kind of calculations can be investigated with Von Neumann stability analysis, and there you will find out that the instability acts by amplifying the short-wavelength Fourier components of the numerical discretization error.

The specific stability condition for the explicit solution of diffusion equation is

In the better method to solve the diffusion equation, the implicit method, we will not solve the numbers $f_{i;j+1}$ in a straightforward way from the numbers $f_{i;j}$. Instead, we use backward time stepping to write the $f_{i;j}$ as a function of the numbers $f_{i-1;j+1}$, $f_{i;j+1}$ and $f_{i+1;j+1}$, as in the equation below:

which represents a linear system of equations. More specifically, it is a tridiagonal system, and in matrix-vector form it reads

for the case of a relatively small mesh $n_x = 7$. So, now we have to solve this tridiagonal system on each timestep, and this take more computation time than the explicit method but has the advantage of making the calculation stable even when $\Delta t$ is not necessarily made much smaller than $\Delta x$.

A code for solving this kind of a linear system can be found from the book “Numerical Recipes for C” or from the corresponding book written for FORTRAN.

Now, let’s use the implicit method to solve a diffusion problem where the x-domain is

$x \in [0,6]$,

and the step sizes are

$\Delta x = 0.01$ and $\Delta t = 0.05$ .

The initial concentration profile is chosen to be

(don’t be confused by the fact that the function is now called C instead of f) and we use a boundary condition that forces the value of $C(x,t)$ at the left endpoint to be $C(0,t)=1$ and that at the right endpoint to be $C(6,t) = 0$. This means that the left boundary is an infinite source of concentration (or heat in the case of conduction problem) and the right boundary is an infinite “sink”. With physical intuition, it is easy to deduce that this kind of system evolves toward a steady state where the value of $C(x)$ decreases linearly from 1 to 0 on the interval $[0,6]$

A C++ code for doing this calculation is shown below.

// This program calculates the time development of a diffusion or heat conduction
// system with implicit finite differencing. The system described is the development of a concentration or temperature field
// between boundary points where it is constrained to stay at different constant values.
// Teemu Isojärvi, Feb 2017

#include
#include

using namespace std;

#define LX 6. // Length of spatial domain
#define NX 600 // Number of lattice points
#define LT 3. // Length of time interval
#define NT 60 // Number of timesteps

#define D 1. // Diffusion coefficient

int main(void)
{

double dx = (double)LX/(double)NX; // Lattice spacing
double dt = (double)LT/(double)NT; // Length of timestep

double c[NX]; // Concentration values at lattice points

double x; // Auxiliary position variable

double kappa = D*dt/(dx*dx); // Auxiliary variable for representing the linear system

double g[NX];
double b;
double q;

double u[NX]; // Vector for storing the solution on each timestep

for(int m = 0; m<NX; m++)
{
x = (double)m*dx;
c[m]=exp(-3*x*x); // Gaussian initial concentration
}

for(int n = 0; n<NT; n++)
{
c[0]=1;
c[NX-1]=0;

u[0]=(c[0]+kappa)/(2*kappa+1);
q = 2*kappa + 1;

for(int m = 1; m < NX; m++) { // First loop for solving the tridiagonal system g[m]=-kappa/q; q = (2*kappa + 1) + kappa*g[m]; u[m]=(c[m]+kappa*u[m-1])/q; } for(int m=(NX-2); m>=0; m--) u[m] -= g[m+1]*u[m+1]; // Second loop

for(int m=0; m<NX; m++) c[m] = u[m]; // Updating the concentration or temperature field

}

for(int m = 0; m<NX; m++)
{
x = (double)m*dx;
cout << x << " " << c[m] << "\n"; // Output with the results at time t = LT
}

return 0;
}

Running this program three times, with domain lenght parameters $L=0.5$, $L=1.0$ and $L=3.0$ and keeping the time step constant, we get data points that can be plotted in the same coordinate system with a graphing program, like below:

Figure 1. Time evolution of a concentration field C(x,t) in a system where the concentration is forced to stay at constant values at the endpoints of the domain.

The simulation seems to proceed as expected, approaching a linearly decreasing function $C(x)$

An equivalent code for FORTRAN is in the next box:

! Calculates the time development of a concentration distribution C(x,t) with implicit
! finite-differencing of a diffusion equation. The boundary condition is that the left boundary is an infinite source
! of solute/heat and the value of C(x) at x=0 stays at constant value 1. The value of C at the right boundary stays zero.
! Therefore, the function C(x) evolves towards a linearly decreasing function.
! Teemu Isojärvi, Feb 2017

PROGRAM MAIN

real :: DX, DT, LX, LT, D, KAPPA,B,Q ! Real variables for discretization and the diffusion constant
integer :: NT,NX ! Number of time and position steps

REAL :: C(600) ! Concentration field as an array of data points, dimension same as value of NX
REAL :: G(600)
REAL :: U(600)

REAL :: X ! Auxiliary variable

INTEGER :: M ! Integer looping variables
INTEGER :: N

LX = 6. ! Length of x-interval
LT = 3.0 ! Length of t-interval
NX = 600 ! Number of lattice points
NT = 300 ! Number of time steps
DX = LX/NX ! Distance between neighboring lattice points
DT = LT/NT ! Length of time step
D = 1. ! Diffusion/heat conduction coefficient
KAPPA = D*DT/(DX*DX)

do M = 1, NX ! Initial values of concentration
X = M*DX
C(M) = EXP(-3*X*X) ! Gaussian initial concentration distribution
end do

do N = 1, NT ! Time stepping loop

C(1)=1
C(NX)=0

U(1)=(C(1) + 2 * KAPPA)/(2 * KAPPA + 1)
Q = 2 * KAPPA + 1

do M = 2, NX ! First loop for solving the tridiagonal system
G(M) = -KAPPA/Q
Q = (2 * KAPPA + 1) + KAPPA * G(M)
U(M) = (C(M) + KAPPA * U(M-1))/Q
end do

do M = (NX-1), 1
U(M) = U(M) - G(M+1) * U(M+1) ! Second loop
end do

do M = 1, NX-1
C(M) = U(M) ! Updating the concentration or temperature field
end do

end do

do M = 1, NX
X = M*DX
print *,X,C(M) ! Print the x and concentration values at data points
end do

END

To test the implicit method with R-Code, let’s solve a problem where the length of the x-domain is 20, the time interval has length 3 and the initial distribution is

to ensure that we can set the boundary conditions $C(0,t)=C(L,t)=0$ (the Gaussian distribution doesn’t have time to spread all the way to the boundaries in a time interval of 3 units). The code for this calculation is shown below:

library(graphics) #load the graphics library needed for plotting
library(limSolve)

lx <- 20.0 #length of the computational domain
lt <- 3. #length of the simulation time interval
nx <- 4000 #number of discrete lattice points
nt <- 300 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell
dt <- lt/nt #length of timestep

D <- 1.0 #diffusion constant

Conc <- c(1:nx) #define a vector to put the x- and t-dependent concentration values in

xaxis <- c(1:nx) #the x values corresponding to the discrete lattice points

kappa <- D*dt/(dx*dx) #a parameter needed in the discretization

offdiagonal <- rep(-kappa, times = nx-1)
ondiagonal <- rep(1+2*kappa, times = nx)

for (i in c(1:nx)) {
Conc[i] <- exp(-2*(i*dx-10)*(i*dx-10)) #a Gaussian initial concentration field

xaxis[i] <- i*dx #calculate the x coordinates of lattice points
}

for (j in c(1:nt)) { #main time stepping loop

sol <- Solve.tridiag(offdiagonal, ondiagonal, offdiagonal, Conc)

for (k in c(1:nx)) {
Conc[k] <- sol[k]
}

if(j %% 3 == 1) { #make plots of C(x) on every third timestep
jpeg(file = paste("plot_",j,".jpg",sep=""))
plot(xaxis,Conc,xlab="position (x)", ylab="concentration",ylim=c(0,2))
title(paste("C(x) at t =",j*dt))
lines(xaxis,Conc)
dev.off()
}
}

An animation of the results can be viewed in this link. If you test the code yourself, remember to install the limSolve package first, by writing the command

install.packages(“limSolve”)

in the R console. If you don’t want to load the video, here are the plots for 3 different values of t:

When solving PDE:s other than the ordinary diffusion equation, the implicit method is often even more necessary that it was in the examples here. For example, when solving the time-development of a quantum wave packet from the time-dependent Schroedinger equation, the solution function doesn’t stay normalized if one tries to do a simple explicit time stepping. The solution of Schroedinger equations with implicit method will be shown in the next post. The TDSE is a complex diffusion equation of the form

where the equation has been simplified by setting the Planck constant to value $2\pi$ and the particle mass to $m=1$.

The 1D diffusion and Schroedinger equations are simple in the sense that the linear system to be solved on each timestep is tridiagonal, with makes the solution considerably faster than is the case with a general linear system. Systems that are not tridiagonal will appear when one wants to solve equations with more than one space coordinate (i.e. x and y instead of just x), or when the equation contains higher than 2nd order derivatives with respect to the position coordinate (this can be demonstrated with the thin-film equation, which is fourth-order with respect to x).