11.2. The method of lines

Our strategy in Section 11.1 was to discretize both the time and space derivatives using finite differences, then rearrange so that we could march the solution forward through time. It was partially effective, but as Demo 11.1.5 shows, not a sure thing, for reasons we look into starting in the next section.

First, though, we want to look at a broader version of the discretization approach. To introduce ideas, let’s use the simpler heat equation, \(u_t = u_{xx}\), as a model. Because boundaries always complicate things, we will start by doing the next best thing to having no boundaries at all: periodic end conditions. Specifically, we will solve the PDE over \(0\le x < 1\) and require at all times that

\[ u(x+1,t)=u(x,t) \quad \text{for all $x$}. \]

This is a little different from simply \(u(1,t)=u(0,t)\), as Fig. 11.2.1 illustrates.

periodic function illustration

Fig. 11.2.1 Left: A function whose values are the same at the endpoints of an interval does not necessarily extend to a smooth periodic function. Right: For a truly periodic function, the function values and all derivatives match at the endpoints of one period.

Semidiscretization

As always, we use \(\hat{u}\) when we specifically refer to the exact solution of the PDE. In order to avoid carrying along redundant information about the function, we use \(x_i = ih\) only for \(i=0,\ldots,m-1\), where \(h=1/m\), and it’s understood that a reference to \(x_m\) is silently translated to one at \(x_0\). More generally, we have the identity

(11.2.1)\[ \hat{u}(x_i,t) = \hat{u}\bigl(x_{(i \bmod{m})},t \bigr)\]

for the exact solution \(\hat{u}\) at any value of \(i\).

Next we define a vector \(\mathbf{u}\) by

\[\begin{split} \mathbf{u}(t) = \begin{bmatrix} u_0(t) \\ u_1(t) \\ \vdots \\ u_n(t) \end{bmatrix}. \end{split}\]

This step is called semidiscretization, since space is discretized but time is not. As in Chapter 10, we will replace \(u_{xx}\) with multiplication of \(\mathbf{u}\) by a differentiation matrix \(\mathbf{D}_{xx}\). The canonical choice is the three-point finite-difference formula (5.4.7), which in light of the periodicity (11.2.1) leads to

(11.2.2)\[\begin{split}\mathbf{D}_{xx} = \frac{1}{h^2} \begin{bmatrix} -2 & 1 & & & 1 \\ 1 & -2 & 1 & & \\ & \ddots & \ddots & \ddots & \\ & & 1 & -2 & 1 \\ 1 & & & 1 & -2 \end{bmatrix}.\end{split}\]

Note well how the first and last rows have elements that “wrap around” from one end of the domain to the other by periodicity. Because we will be using this matrix quite a lot, we create Function 11.2.1 to compute it, as well as the corresponding second-order first derivative matrix \(\mathbf{D}_x\) for periodic end conditions.

Function 11.2.1 :  diffper

Differentiation matrices for periodic end conditions

 1"""
 2    diffper(n,xspan)
 3
 4Construct 2nd-order differentiation matrices for functions with
 5periodic end conditions, using `n` unique nodes in the interval
 6`xspan`. Returns a vector of nodes and the matrices for the first
 7and second derivatives.
 8"""
 9function diffper(n,xspan)
10    a,b = xspan
11    h = (b-a)/n
12    x = @. a + h*(0:n-1)   # nodes, omitting the repeated data
13
14    # Construct Dx by diagonals, then correct the corners.
15    dp = fill(0.5/h,n-1)        # superdiagonal
16    dm = fill(-0.5/h,n-1)       # subdiagonal
17    Dx = diagm(-1=>dm,1=>dp)
18    Dx[1,n] = -1/(2*h)
19    Dx[n,1] = 1/(2*h)
20
21    # Construct Dxx by diagonals, then correct the corners.
22    d0 =  fill(-2/h^2,n)        # main diagonal
23    dp =  ones(n-1)/h^2         # superdiagonal and subdiagonal
24    Dxx = diagm(-1=>dp,0=>d0,1=>dp)
25    Dxx[1,n] = 1/(h^2)
26    Dxx[n,1] = 1/(h^2)
27
28    return x,Dx,Dxx
29end

The PDE \(u_t=u_{xx}\) is now approximated by the semidiscrete problem

(11.2.3)\[ \frac{d \mathbf{u}(t)}{d t} = \mathbf{D}_{xx} \mathbf{u}(t),\]

which is simply a linear, constant-coefficient system of ordinary differential equations. Given the initial values \(\mathbf{u}(0)\) obtained from \(u(x_i,0)\), we have an initial-value problem that we already know how to solve!

Semidiscretization is often called the method of lines. Despite the name, it is not exactly a single method because both space and time discretizations have to be specified in order to get a concrete algorithm. The key concept is the separation of those two discretizations, and in that way, it’s related to separation of variables in analytic methods for the heat equation.

Example 11.2.2

Suppose we solve (11.2.3) using the Euler IVP integrator (6.2.3) from Section 6.2 (and also AB1 from Section 6.6). We select a time step \(\tau\) and discrete times \(t_j=j\tau\), \(j=0,1,\ldots,n\). We can discretize the vector \(\mathbf{u}\) in time as well to get a sequence \(\mathbf{u}_j \approx \mathbf{u}(t_j)\) for varying \(j\). (Remember the distinction in notation between \(\mathbf{u}_j\), which is a vector, and \(u_j\), which is a single element of a vector.)

Thus, a fully discrete method for the heat equation is

(11.2.4)\[\mathbf{u}_{j+1} = \mathbf{u}_j + \tau ( \mathbf{D}_{xx} \mathbf{u}_j) = (\mathbf{I} + \tau \mathbf{D}_{xx} ) \mathbf{u}_j.\]
Demo 11.2.3

Let’s implement the method of Example 11.2.2 with second-order space semidiscretization.

m = 100
x,Dx,Dxx = FNC.diffper(m,[0,1]);

tfinal = 0.16;  n = 2400;  
τ = tfinal/n;  t = τ*(0:n);

Next we set an initial condition. It isn’t mathematically periodic, but the end values and derivatives are so small that for numerical purposes it may as well be.

U = zeros(m,n+1);
U[:,1] = @. exp( -60*(x-0.5)^2 )
plot(x,U[:,1],xaxis=(L"x"),yaxis=(L"u(x,0)"),
    title="Initial condition")
Initial condition

The Euler time stepping simply multiplies by the constant matrix in (11.2.4) at each time step. Since that matrix is sparse, we will declare it as such, even though the run-time savings may not be detectable for this small value of \(m\).

A = sparse(I+τ*Dxx)
for j in 1:n
    U[:,j+1] = A*U[:,j]
end

idx = [1,21,41,61]
times = round.(t[idx],digits=4)
label = reshape(["t = $t" for t in times],1,length(idx))
plot(x,U[:,idx];label,
    title="Heat equation by forward Euler",legend=:topleft,  
    xaxis=(L"x"),yaxis=(L"u(x,0)",[0,1]))
Heat equation by forward Euler

Things seem to start well, with the initial peak widening and shrinking. But then there is a nonphysical growth in the solution.

anim = @animate for j in 1:101
    plot(x,U[:,j],label=@sprintf("t=%.5f",t[j]),
    xaxis=(L"x"),yaxis=(L"u(x,t)",[-1,3]),dpi=100,
    title="Heat equation by forward Euler")
end
mp4(anim,"diffusionFE.mp4")
┌ Info: Saved animation to 
│   fn = /Users/driscoll/repos/fnc-julia/diffusion/diffusionFE.mp4
└ @ Plots /Users/driscoll/.julia/packages/Plots/8K4be/src/animation.jl:114

The growth in norm is exponential in time.

M = vec( maximum(abs,U,dims=1) )   
plot(t[1:1000],M[1:1000],
    xaxis=(L"t"), yaxis=(:log10,L"\max_x |u(x,t)|"),
    title="Nonphysical growth") 
Nonphysical growth

The method in Example 11.2.2 and Demo 11.2.3 is essentially the same one we used for the Black–Scholes equation in Section 11.1. By changing the time integrator, we can get much better results.

Example 11.2.4

An alternative time discretization of (11.2.3) is to use the backward Euler (AM1) method, resulting in

(11.2.5)\[\begin{split}\begin{split} \mathbf{u}_{j+1} &= \mathbf{u}_j + \tau (\mathbf{D}_{xx} \mathbf{u}_{j+1})\\ (\mathbf{I} - \tau \mathbf{D}_{xx}) \mathbf{u}_{j+1} &= \mathbf{u}_j. \end{split}\end{split}\]

Because backward Euler is an implicit method, a linear system must be solved for \(\mathbf{u}_{j+1}\) at each time step.

Demo 11.2.5

Now we apply backward Euler to the heat equation. We will reuse the setup from Demo 11.2.3. Since the matrix in (11.2.5) never changes during the time stepping, we do the necessary LU factorization only once.

B = sparse(I-τ*Dxx)
factor = lu(B)
for j in 1:n
    U[:,j+1] = factor\U[:,j]
end
idx = 1:600:n+1
times = round.(t[idx],digits=4)
label = reshape(["t = $t" for t in times],1,length(idx))
plot(x,U[:,idx];label,
    title="Heat equation by backward Euler",legend=:topleft,  
    xaxis=(L"x"),yaxis=(L"u(x,0)",[0,1]))
Heat equation by backward Euler
anim = @animate for j in 1:20:n+1
    plot(x,U[:,j],label=@sprintf("t=%.5f",t[j]),
    xaxis=(L"x"),yaxis=(L"u(x,t)",[0,1]),dpi=100,
    title="Heat equation by backward Euler")
end
mp4(anim,"diffusionBE.mp4")
┌ Info: Saved animation to 
│   fn = /Users/driscoll/repos/fnc-julia/diffusion/diffusionBE.mp4
└ @ Plots /Users/driscoll/.julia/packages/Plots/8K4be/src/animation.jl:114

This solution looks physically plausible, as the large concentration in the center diffuses outward until the solution is essentially constant. Observe that the solution remains periodic in space for all time.

Demo 11.2.5 suggests that implicit time stepping methods have an important role in diffusion. We will analyze the reason in the next few sections.

Black-box IVP solvers

Instead of coding one of the Runge–Kutta or multistep formulas directly for a method of lines solution, we could use any of the IVP solvers from Chapter 6, or a solver from the DifferentialEquations package, to solve the ODE initial-value problem (11.2.3).

Demo 11.2.6

We set up the semidiscretization and initial condition in \(x\) just as before.

m = 100
x,Dx,Dxx = FNC.diffper(m,[0,1])
u0 = @. exp(-60*(x-0.5)^2);

Now, however, we apply Function 6.5.2 (rk23) to the initial-value problem \(\mathbf{u}'=\mathbf{D}_{xx}\mathbf{u}\).

tfinal = 0.25
ODE = (u,p,t) -> Dxx*u;  
IVP = ODEProblem(ODE,u0,(0,tfinal))

t,u = FNC.rk23(IVP,1e-5);

We check that the resulting solution looks realistic.

plt = plot(title="Heat equation by rk23",legend=:topleft,  
    xaxis=(L"x"),yaxis=(L"u(x,0)",[0,1]))
for idx in 1:600:n+1
    plot!(x,u[idx],label="t = $(round.(t[idx],digits=4))")
end
plt
Heat equation by rk23
anim = @animate for j in 1:20:1600
    plot(x,u[j],label=@sprintf("t=%.4f",t[j]),
      xaxis=(L"x"), yaxis=(L"u(x,t)",[0,1]),dpi=100,
      title="Heat equation by rk23")
end
mp4(anim,"diffusionRK23.mp4")
┌ Info: Saved animation to 
│   fn = /Users/driscoll/repos/fnc-julia/diffusion/diffusionRK23.mp4
└ @ Plots /Users/driscoll/.julia/packages/Plots/8K4be/src/animation.jl:114

The solution appears to be correct. But the number of time steps that were selected automatically is surprisingly large, considering how smoothly the solution changes.

println("Number of time steps for rk23: $(length(t)-1)")
Number of time steps for rk23: 3975

Now we apply a solver from DifferentialEquations.

u = solve(IVP,Rodas4P());
println("Number of time steps for Rodas4P: $(length(u.t)-1)")
Number of time steps for Rodas4P: 23

The number of steps selected is reduced by a factor of more than 100!

The adaptive time integrators can all produce solutions. But, as seen in Demo 11.2.6, they are not equivalent in every important sense. Whether we choose to implement a method directly with a fixed step size, or automatically with adaptation, there is something crucial to understand about the semidiscrete problem (11.2.3) that will occupy our attention in the next two sections.

Exercises

  1. ⌨ Revisit Demo 11.2.3. For each \(m=20,30,\dots,120\) points in space, let \(n=20,30,40,\dots\) in turn until you reach the smallest \(n\) such that the numerical solution remains bounded above by 2 for all time; call this value \(N(m)\). Make a log-log plot of \(N(m)\) as a function of \(m\). If you suppose that \(N=O(m^p)\) for a simple rational number \(p\), what is a reasonable hypothesis for \(p\)?

  2. In Demo 11.2.6, as \(t\to \infty\) the solution \(u(x,t)\) approaches a value that is constant in both space and time.

    (a) ⌨ Set \(m=400\) and use Rodas4P, as shown in Demo 11.2.6, to find this constant value to at least eight digits of accuracy.

    (b) ✍ Prove that \(\int_0^1 u(x,t) \,dx\) is constant in time.

    (c) ⌨ Use Function 5.6.4 on the initial condition function, and compare to the result of part (a).

  3. ✍ Apply the trapezoid IVP formula (AM2) to the semidiscretization (11.2.3) and derive what is known as the Crank–Nicolson method:

    (11.2.6)\[(\mathbf{I} - \tfrac{1}{2}\tau \mathbf{D}_{xx}) \mathbf{u}_{j+1} = (\mathbf{I} + \tfrac{1}{2}\tau \mathbf{D}_{xx}) \mathbf{u}_{j}.\]

    Note that each side of the method is evaluated at a different time level.

  4. ⌨ Repeat Demo 11.2.5 using the Crank–Nicolson method (11.2.6). Then try for \(n=240\) as well, which uses a time step ten times larger than before. Does the solution remain stable?

  5. The PDE \(u_t = 2u + u_{xx}\) combines growth with diffusion.

    (a) ✍ Derive an equation analogous to (11.2.5) that combines second-order semidiscretization in space with the backward Euler solver in time.

    (b) ⌨ Apply your formula from part (a) to solve this PDE with periodic boundary conditions for the same initial condition as in Demo 11.2.5. Use \(m=200\) points in space and \(n=1000\) time levels. Plot the solution on one graph at times \(t=0,0.04,0.08,\ldots,0.2\), or animate the solution over \(0\le t \le 0.2\).

  6. ✍ In this problem, you will analyze the convergence of the explicit method given by (11.2.4). Recall that the discrete approximation \(u_{i,j}\) approximates the solution at \(x_i\) and \(t_j\).

    (a) Write the method in scalar form as

    \[ u_{i,j+1} = (1-2\lambda) u_{i,j} + \lambda u_{i+1,j} + \lambda u_{i-1,j}, \]

    where \(\lambda = \tau/h^2>0\).

    (b) Taylor series of the exact solution \(\hat{u}\) imply that

    \[\begin{align*} \hat{u}_{i,j+1} &= u_{i,j} + \frac{\partial \hat{u}}{\partial t} (x_i,t_j) \tau + O(\tau^2),\\ % \frac{\partial^2 u}{\partial t^2} (x_i,\bar{t}) \frac{\tau^2}{2} \hat{u}_{i\pm1,j} &= \hat{u}_{i,j} \pm \frac{\partial \hat{u}}{\partial x} (x_i,t_j) h + \frac{\partial^2 \hat{u}}{\partial x^2} (x_i,t_j) \frac{h^2}{2} \pm \frac{\partial^3 \hat{u}}{\partial x^3} (x_i,t_j) \frac{h^3}{6}+ O(h^4). %\frac{\partial^4 u}{\partial x^4} (\bar{x}_\pm,t_j) \frac{h^4}{24}. \end{align*}\]

    Use these to show that

    \[\begin{align*} \hat{u}_{i,j+1} & = \left[ (1-2\lambda) \hat{u}_{i,j} + \lambda \hat{u}_{i+1,j} + \lambda \hat{u}_{i-1,j}\right] + O\Bigl(\tau^2+h^2 \Bigr)\\ &= F\left( \lambda,\hat{u}_{i,j}, \hat{u}_{i+1,j} , \hat{u}_{i-1,j}\right) + O\Bigl(\tau^2+h^2\Bigr). \end{align*}\]

    (The last line should be considered a definition of the function \(F\).)

    (c) The numerical solution satisfies

    \[ u_{i,j+1}=F\bigl( \lambda,u_{i,j}, u_{i+1,j} , u_{i-1,j}\bigr) \]

    exactly. Using this fact, subtract \(u_{i,j+1}\) from both sides of the last line in part (b) to show that

    \[ e_{i,j+1} = F\left( \lambda,e_{i,j}, e_{i+1,j} ,e_{i-1,j}\right) + O\Bigl(\tau^2+h^2\Bigr), \]

    where \(e_{i,j}=\hat{u}_{i,j}-u_{i,j}\) is the error in the numerical solution for all \(i\) and \(j\) .

    (d) Define \(E_j\) as the maximum of \(|e_{i,j}|\) over all values of \(i\), and use the result of part (c) to show that if \(\lambda<1/2\) is kept fixed as \(h\) and \(\tau\) approach zero, then for sufficiently small \(\tau\) and \(h\),

    \[ E_{j+1} = E_{j} + O\Bigl(\tau^2+h^2\Bigr) \le E_{j} + K_j\bigl(\tau^2+h^2\bigr) \]

    for a positive \(K_j\) independent of \(\tau\) and \(h\).

    (e) If the initial conditions are exact, then \(E_0=0\). Use this to show finally that if the \(K_j\) are bounded above and \(\lambda<1/2\) is kept fixed, then \(E_n = O(\tau)\) as \(\tau\to 0\).