## Numerical solution of PDE:s, Part 10: The thin-film equation

Earlier, I showed how to solve the 1D and 2D versions of the complex Ginzburg-Landau equation, which is an example of a nonlinear partial differential equation, and which had to be linearized for solution with implicit differencing, meaning that the matrix in the linear system was different on each timestep.

Another nonlinear PDE is the so-called thin film equation, which in 2D form reads

Here the function $h(x,y,t)$ describes the local thickness of a film of viscous liquid located on top of a solid surface described by the xy-plane. The parameter $\gamma$ is the surface tension of the liquid-gas interface and $\mu$ is the viscosity of the liquid.

An unusual thing about this equation is that it’s fourth order in the spatial coordinates, while most equations in physics are second order DE:s. Some equations of continuum mechanics describing elastic bending of cylinders and plates are also fourth order, so this is not the only example.

In many cases, a corresponding equation with only one spatial coordinate is enough for describing thin film physics, and then the graph of the solution can be though of as depicting an intersection of the film at a single value of y-coordinate.

When discretizing this equation, we must note that the factor $h^3$ has to be treated explicitly to get a linear system of equations, just like what had to be done with the $|A|^2$ in the CGLE. Also, we will set $\gamma/(3\mu ) = 1$ to make the equation dimensionless. One correct way to discretize this equation leads to the system

where the two index object $\alpha_{j}^{i}$ is

Note that now the linear system is not tridiagonal, but heptadiagonal due to the higher order derivatives. Solution of the equation with this differencing scheme and a Gaussian initial condition $h(x,0)$ is done with the following R language code.

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

lx <- 10 #length of the computational domain
lt <- 10 #length of the simulation time interval
nx <- 150 #number of discrete lattice points
nt <- 150 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell
dt <- lt/nt #length of timestep

psi = c(1:nx) #array for the function h values
sol = c(1:nx)

kappa = dt/(4*dx*dx*dx*dx)

for(j in c(1:nx)) {
psi[j] = exp(-(j*dx-5)*(j*dx-5))
sol[j] = psi[j]
}

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

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

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

for(j in c(1:nx)) {
alpha[j] = kappa*psi[j]*psi[j]*psi[j]
}

for(j in c(1:nx)) {
for(k in c(1:nx)) {
A[j,k]=0
if(j==k) {
if(j!=nx && j!=1) {
A[j,k] = 1 + 2*alpha[j+1] + 2*alpha[j-1] #diagonal elements
}
if(j==1) {
A[j,k] = 1 + 2*alpha[j+1]
}
if(j==nx) {
A[j,k] = 1 + 2*alpha[j-1]
}
}

if(j==k+1 && j!=1) {
A[j,k] = -alpha[j-1] #off-diagonal elements
}

if(j==k-1 && j!=nx) {
A[j,k] = -alpha[j+1]
}

if(j==k+2 && j<nx) {
A[j,k] = -2*alpha[j+1]
}

if(j==k-2 && j>1) {
A[j,k] = -2*alpha[j-1]
}

if(j==k+3 && j<nx) {
A[j,k] = alpha[j+1]
}

if(j==k-3 && j>1) {
A[j,k] = alpha[j-1]
}

}
}

for(l in c(1:nx)) {
psi[l] = sol[l]
}
sol <- solve(A,psi) #solve the system of equations

jpeg(file = paste("plot_",m,".jpg",sep=""))
plot(xaxis,sol,xlab = "position (x)",ylab="h(x,t)",ylim=c(0,1),pch='.')
title(paste("h(x,t) at t = ",round(m*dt,digits=2)))
lines(xaxis,sol)
dev.off()

}

dev.off()

An animation of the solution looks like this.

The solutions of this equation have the property that the graph settles into the shape of a downward opening parabola when time proceeds. A problem with this is that the contact angle, in which the liquid surface approaches the solid surface (x-axis) at the final equilibrium, can be anything between 0 and 90 degrees depending on the relative width and height of the initial Gaussian. In a real liquid-solid system, the final contact angle depends on the surface tensions of both the liquid-solid and the liquid-gas interfaces, as described by the Young equation.

To create an equation that can model equilibrium contact angles appropriately, we add a disjoining pressure term $\Pi (h)$ in the equation, as here:

One form of the $\Pi$-term that works is

where the $h_*$ is a precursor film thickness.

The idea behind the precursor film is that even when we have a liquid drop or puddle surrounded by apparently dry solid surface, there is actually a very thin adsorbed layer of liquid molecules of that dry area (the molecules get there by evaporating from the liquid surface and reattaching on the solid). So, in a simulation where we include the disjoining pressure, we need to use an initial condition that is a Gaussian with an added constant equal to the precursor thickness:

$h(x,0) = \exp \left[-b(x-x_0 )^2 \right] + h_*$

In here we will set the values n = 5 and m = 2 in the disjoining pressure term, and set the precursor film thickness into the value 0.01. The disjoining pressure term can be treated explicitly at the same time as we use implicit differencing for the rest of the equation – we subtract, on the RHS of the discretized equation a term $D_{j}^i$,

defined by

where

and $\Pi_{j}^i$ is the disjoining pressure evaluated at the discrete points. Note the use of a central finite difference instead of a one-sided difference when calculating the derivatives – doing otherwise is likely to make the simulation crash. The term $D_{j}^{i}$ only affects the right-hand side vector of the linear system $\mathbf{Ax} = \mathbf{b}$ that we solve on each timestep. A code that solves the new equation for the pre-factor value $B=0.1$ is shown next.

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

lx <- 10 #length of the computational domain
lt <- 5 #length of the simulation time interval
nx <- 150 #number of discrete lattice points
nt <- 5000 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell
dt <- lt/nt #length of timestep

prec = 0.01 #precursor film thickness

psi = c(1:nx) #array for the function h values
sol = c(1:nx)

disjoin1 = c(1:nx)
disjoin2 = c(1:nx)
disjoin3 = c(1:nx)

kappa = dt/(4*dx*dx*dx*dx)

for(j in c(1:nx)) {
psi[j] = exp(-(j*dx-5)*(j*dx-5))+prec
sol[j] = psi[j]
}

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

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

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

disjoin1[1] = 0
disjoin1[nx] = 0
for(j in c(1:nx)) {
disjoin1[j] = 0.1*((prec/psi[j])^5 - (prec/psi[j])^2)
}

for(j in c(2:(nx-2))) {
disjoin2[j] = psi[j]*psi[j]*psi[j]*(disjoin1[j+1] - disjoin1[j-1])/(2*dx)
}

for(j in c(2:(nx-2))) {
disjoin3[j] = (disjoin2[j+1]-disjoin2[j-1])/(2*dx)
}

disjoin3[1] = 0
disjoin3[2] = 0
disjoin3[nx-1] = 0
disjoin3[nx] = 0

for(j in c(1:nx)) {
alpha[j] = kappa*psi[j]*psi[j]*psi[j]
}

for(j in c(1:nx)) {
for(k in c(1:nx)) {
A[j,k]=0
if(j==k) {
if(j!=nx && j!=1) {
A[j,k] = 1 + 2*alpha[j+1] + 2*alpha[j-1] #diagonal elements
}
if(j==1) {
A[j,k] = 1 + 2*alpha[j+1]
}
if(j==nx) {
A[j,k] = 1 + 2*alpha[j-1]
}
}

if(j==k+1 && j!=1) {
A[j,k] = -alpha[j-1] #off-diagonal elements
}

if(j==k-1 && j!=nx) {
A[j,k] = -alpha[j+1]
}

if(j==k+2 && j<nx) {
A[j,k] = -2*alpha[j+1]
}

if(j==k-2 && j>1) {
A[j,k] = -2*alpha[j-1]
}

if(j==k+3 && j<nx) {
A[j,k] = alpha[j+1]
}

if(j==k-3 && j>1) {
A[j,k] = alpha[j-1]
}

}
}

for(l in c(1:nx)) {
psi[l] = sol[l]
}
sol <- solve(A,psi-disjoin3) #solve the system of equations
for(l in c((nx-20):nx)) { # remove the risk of boundary effects on the right end of the domain
psi[l]=prec
sol[l]=prec
}

if(m%%10 == 0) {
jpeg(file = paste("plot_",m,".jpg",sep=""))
plot(xaxis,sol,xlab = "position (x)",ylab="h(x,t)",ylim=c(0,1),pch='.')
title(paste("h(x,t) at t = ",round(m*dt,digits=2)))
lines(xaxis,sol)
dev.off()
}

}

In the next animation, the solution curves for pre-factor values $B=0.1$ (red curve) and $B=0.01$ (black curve) are shown in the same graph.

In the animation, it is apparent that a larger value of $B$ leads to a larger contact angle at equilibrium. Actually, it can be shown that the contact angle $\theta$ depends on the values of the parameters as

More information about the thin film equation and its solutions can be found on the Wiki page here, and on the NJIT department of applied mathematics homepage. When the effects of gravitation or surface tension gradients are added in the TFE, many kinds of interesting pattern formation effects can happen just like in our previous example of a nonlinear PDE, the Ginzburg-Landau equation. If the liquid film consists of a mixture of many, possibly volatile liquids, complicated multiphysics problems involving fluid dynamics, evaporation, heat transfer and chemical kinetics, all at the same time, are obtained.

## Numerical solution of PDE:s, Part 9: 2D Ginzburg-Landau equation

In an earlier post, I described the 1-dimensional Ginzburg-Landau equation and showed how it can be linearized and solved with a implicit differencing scheme. The most interesting feature of the solutions was the appearance of seemingly random oscillations. A similar solution method is possible for the 2d version of the equation:

where again $\alpha$ and $\beta$ are real valued constants.

An R language code for solving this with parameter values $\alpha = 0$, $\beta = 1.5$ and an initial state $A(x,0)$ which is a mixture of 2d plane waves is shown below.

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

lx <- 80.0 #length of the computational domain in x-direction
ly <- 80.0 #length of the computational domain in y-direction
lt <- 60.0 #length of the simulation time interval
nx <- 50 #number of discrete lattice points in x-direction
ny <- 50 #number of discrete lattice points in y-direction
nt <- 240 #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

a <- 0
b <- 1.5

kappa1 = dt*(1+1i*a)/dx/dx
kappa2 = dt*(1+1i*b)

C = c(1:(nx*ny))
Cu = c(1:(nx*ny))
A2d = 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

A = matrix(nrow=(nx*ny),ncol=(nx*ny))
IP = matrix(nrow=4*nx,ncol=4*nx)

for (i in c(1:ny)) {
for (j in c(1:nx)) {
A2d[i,j] <- 0.01*exp(1i*5.21*i+1i*10.331*j)+0.01*exp(1i*15.71*i+1i*17.831*j)
}
}

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

for(i in c(1:ny)) {
for(j in c(1:nx)) {
C[(i-1)*nx+j] <- A2d[i,j]

}
}

for(i in c(1:(nx*ny))) {
for(j in c(1:(nx*ny))) {
A[i,j] <- 0
if(i==j && j!=1 && j!=nx && i!=1 && i!=ny) A[i,j] <- 1+2*kappa1+kappa2*abs(C[j])*abs(C[j]) - dt
if(i==j && (j==1 || j==nx) && i!=1 && i!=ny) A[i,j] <- 1+2*kappa1+kappa2*abs(C[j])*abs(C[j]) - dt
if(i==j && j!=1 && j!=nx && (i==1 || i==ny)) A[i,j] <- 1+2*kappa1+kappa2*abs(C[j])*abs(C[j]) - dt
if(i==j && (j==1 || j==nx) && (i==1 || i==ny)) A[i,j] <- 1+2*kappa1+kappa2*abs(C[j])*abs(C[j]) - dt
if(j==i+1 && (i%%nx != 0)) A[i,j] <- -kappa1
if(j==i-1 && (i%%nx != 1)) A[i,j] <- -kappa1
if(j==i+nx) A[i,j] <- -kappa1
if(j==i-nx) A[i,j] <- -kappa1
}
}

Cu <- solve(A,C)

for(i in c(1:ny)) {
for(j in c(1:nx)) {
if(i==1) Cu[(i-1)*nx+j]=Cu[i*nx+j]
if(i==ny) Cu[(i-1)*nx+j]=Cu[(i-2)*nx+j]
if(j==1) Cu[(i-1)*nx+j]=Cu[(i-1)*nx+j+1]
if(j==nx) Cu[(i-1)*nx+j]=Cu[(i-1)*nx+j-1]
}
}

for(i in c(1:ny)) {
for(j in c(1:nx)) {
A2d[i,j] <- Cu[(i-1)*nx+j]
}
}

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] = A2d[l,m]
IP[4*l-2,4*m-3] = A2d[l,m]+0.25*(A2d[l+1,m]-A2d[l,m])
IP[4*l-1,4*m-3] = A2d[l,m]+0.5*(A2d[l+1,m]-A2d[l,m])
IP[4*l,4*m-3] = A2d[l,m]+0.75*(A2d[l+1,m]-A2d[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])
}
}

#make plots of C(x,y) on every third timestep
jpeg(file = paste("plot_",k,".jpg",sep=""))
image(Re(IP),zlim=c(-3,3))
title(paste("Real part of solution A(x,y,t)",k*dt))
dev.off()

}

The code produces 2d plots of the real part of the solution on each timestep, and in the video shown below they have been combined into an animation.

In the animation we see the appearance of spiral patterns typical for these values of parameters $\alpha,\beta$. Other values of the parameters
produce different kinds of patterns, as is described in this link.

## Numerical solution of PDE:s, Part 8: Complex Ginzburg-Landau Equation

In the previous numerical solution posts, I described linear equations like diffusion equation and the Schrödinger equation, and how they can be solved by (implicit or explicit) finite differencing. The idea of the implicit methods was to convert the equation into a linear system of equations, from which the function values on a discrete mesh could be calculated.

Saying that these equations were linear means that they can be written as

where the linear differential operator, containing space and time derivatives, is acting on the function and producing “something” (usually zero but in the case of source terms/inhomogeneity something nonzero).

As a first example of a nonlinear PDE, let’s consider the complex Ginzburg-Landau equation (CGLE), which reads:

Here the $\alpha$ and $\beta$ are real parameters and $i$ is the imaginary unit. Applying an implicit differencing on this may seem to result in a system of equations

but this is not a linear system because of the $|A_{i}^{j+1}|^2$, so we cannot solve the problem in this way by using linear algebra.

The trick to solve this is to linearize the system, by evaluating the $|A|^2$ at timestep $j$ and the rest of the quantities at timestep $j+1$, producing the system

which is now a linear system w.r.t. to the variables evaluated at timestep $j+1$ (the matrix for solving “$A^{j+1}$“:s has diagonal elements that depend of “$A^j$“:s). A more sophisticated method would do several iterations to approximate the values of $A(x,t)$ between the timesteps $j$ and $j+1$.

An R code that solves the equation for a domain $x\in [0,100]$, $t\in [0,150]$, using discrete steps $\Delta x = 0.66$, $\Delta t = 0.33$ , initial state $A(x,0) = 0.1e^{2ix}$ and values $\alpha=3$ and $\beta = -2$, is shown here.

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

lx <- 100 #length of the computational domain
lt <- 150 #length of the simulation time interval
nx <- 150 #number of discrete lattice points
nt <- 450 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell
dt <- lt/nt #length of timestep

a <- 3
b <- -2

kappa1 = dt*(1+1i*a)/dx/dx #an element needed for the matrices
kappa2 = dt*(1+1i*b)

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

for(j in c(1:nx)) {
psi[j] = 0.1*exp(2i*j*dx)
sol[j] = psi[j]
}

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

IPxaxis <- c(1:(4*nx))*dx/4
IPtaxis <- c(1:(4*nt))*dt/4

sol_plot = matrix(nrow=nt,ncol=nx)

A = matrix(nrow=nx,ncol=nx) #matrix for forward time evolution
IP = matrix(nrow = 4*nt, ncol=4*nx)

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

for(j in c(1:nx)) {
for(k in c(1:nx)) {
A[j,k]=0
if(j==k) {
A[j,k] = 1 + 2*kappa1 + kappa2*abs(sol[j])*abs(sol[j]) – dt #diagonal elements
}
if((j==k+1) || (j==k-1)) {
A[j,k] = -kappa1 #off-diagonal elements
}
}
}

for(l in c(1:nx)) {
psi[l] = sol[l]
}
sol <- solve(A,psi) #solve the system of equations

for (l in c(1:nx)) {
sol_plot[m,l] <- Re(sol[l])
}

jpeg(file = paste(“plot_”,m,”.jpg”,sep=””))
plot(xaxis,Im(sol),xlab = “position (x)”,ylab=”Im[A(x,t)]”,ylim=c(-4,4),pch=’.’)
title(paste(“Im[A(x,t)] at t = “,round(m*dt,digits=2)))
lines(xaxis,Im(sol))
dev.off()

}

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

for(l in c(1:(4*nt))) {
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 = “2dplot.jpg”)
image(IPtaxis,IPxaxis,IP,xlab = “t-axis”,ylab=”x-axis”,zlim=c(-2,2))
dev.off()

Plotting the real part of the resulting function $A(x,t)$ at several values of $t$, we see that the solution initially doesn’t do much anything, but at some point a “phase turbulence” sets in starting from the ends of the x-domain and after that the function evolves in a very random way, without following any clear pattern (unlike the spreading mass/temperature distributions, traveling waves or scattering wavepackets in the case of the common linear PDE:s).

An animation of the solution is shown below.

This kind of chaos is typical of nonlinear systems, be them point mass systems with nonlinear forces between mass points or field systems with nonlinear field equations such as the CGLE here. Note that the solution of this equation is a bit too heavy of a calculation to do just for the purpose of creating random numbers, so for that end other methods such as Perlin’s noise are used.

The 2D color plot of the real part of the solution, plotted in the xt-plane, looks like this:

More plots of the solutions for different values of parameters can be found in this article.

It should be noted that, Wolfram Mathematica’s “NDSolve” function can’t usually solve nonlinear PDE:s correctly, despite usually working properly in the case of linear PDE:s. Some other commercial math programs such as Comsol Multiphysics may work better when solving nonlinear problems, at least to my experience.

So, here was the basic idea of how nonlinear PDE:s are solved by linearization, and what kind of things are possible in the behavior of their solutions. In the next PDE post I will show how to solve the thin-film equation, about which I actually wrote my master’s thesis in 2013, and which doesn’t usually behave chaotically unlike the CGLE (but can be made to do so by adding suitable terms).