# Stability issue with MethodOfLines

Hi!
I’m trying to solve a (I would think) simple PDE with MethodOfLines.jl
The PDE in question is:

\partial_\tau g(τ,x) = \frac{1}{2}σ^2\partial^2_{x} g(τ,x)+ λ(\bar{x} - x)*\partial_x g(τ,x) - r*g(τ,x)
g(0,x) = e^{-kx}
x\in[-2,2]
where \sigma,\lambda,\bar{x},r,k are parameters.
If one wanted an analytical solution I’m pretty sure that by using Feynman-Kac and the formula for the expectation of a lognormal variable one can get it, but this is not why I’m asking your help.

In order to solve this problem numerically, I’m imposing some Neumann condition
\partial_x g(\tau,0)=\partial_x g(\tau,2)=0

If I solve this problem (I tried many solvers from ODE Solvers · DifferentialEquations.jl, like Tsit5(), Sundials.CVODE_BDF(), TRBDF2(), etc. ) on \tau\in [0,1]
I get some wigglines in the solution that then makes the whole solution unstable.
In the pic you can see an example where I plotted the cross section of the solution for \tau = 0.12

What am I doing wrong? Is it a problem of using an adaptive quadrature method?
Here a minimal code that reproduces the issue:

using MethodOfLines, OrdinaryDiffEq, Sundials, ModelingToolkit, Plots
import ModelingToolkit: Interval

τ_min= 0.
τ_max = 1
τ_span = (τ_min,τ_max)

x_min = -2.
x_max = 2.

dx = 0.01

σ = 0.03
λ = 0.1
xb = 1.
r = 0.01
k = 1.
h(x) = exp.(-k*x)

@parameters τ x
@variables g(..)

Dx = Differential(x)
Dxx = Differential(x)^2
Dτ = Differential(τ)

# PDE
eq  = 0 ~ 0.5*σ^2*Dxx(g(τ,x)) + λ*(xb - x)*Dx(g(τ,x)) - Dτ(g(τ,x)) - r*g(τ,x)

# Initial and boundary conditions
bcs = [g(τ_min,x) ~ h(x),
Dx(g(τ,x_max)) ~ 0,
Dx(g(τ,x_min)) ~ 0]

# Space and time domains
domains = [τ ∈ Interval(τ_min,τ_max),
x ∈ Interval(x_min,x_max)]

@named pde_system = PDESystem(eq,bcs,domains,[τ,x],[g(τ, x)])

discretization = MOLFiniteDifference([x=>dx], τ)
prob = discretize(pde_system,discretization)
sol = MethodOfLines.solve(prob, Sundials.CVODE_BDF())
grid = get_discrete(pde_system, discretization)
xs = grid[x]
τs = sol.t
# Retrieve shaped solution
g_lines = [map(d -> sol[d][i], grid[g(τ, x)]) for i in 1:length(sol.t)]

plot(xs,g_lines[20,:])


That does look like quite a nasty instability.

It seems to me that your initial and boundary conditions don’t match unless you suppose that there is a discontinuity in the first derivative at the left hand side (where the instability is occuring). Have you tried the same problem with initial conditions that match the boundary conditions?

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Mmm, something curious is going on here. I tried what @JackDevine suggested using for example:

h(x) = (x>= x_min+5*dx).*exp.(-k*x)+(1-(x>= x_min+5*dx)).*exp.(-k*(x_min+5*dx))



which has a kink, but satisfies the incriminated boundary condition on the left limit of the domain. Nothing really changed.
On the other hand, if I try to change the value of xb, i.e. where the coefficient of the first derivative changes sign to 0 from 1 (closer to the left boundary) then everything runs smoothly.

Can you try a smoother version ? MOL is basically finite differences so I expect big gradients if your initial condition is not smooth.

I believe MOL uses the wrong sign when discretizing the first derivate. It should generate the appropriate upwind operator, but gets confused by (xb-x) I reckon.

When I turn the sign around

eq  = Dτ(g(τ,x)) ~ 0.5*σ^2*Dxx(g(τ,x)) - λ*(x - xb)*Dx(g(τ,x)) - r*g(τ,x)


the instability is gone (I’ve extended the integration time to 5)

I have also used initial conditions that are commensurable with the boundary conditions.

h(x) = 1.0 + cos(π*(x-x_min)/(x_max-x_min))


Note that these BCs alone do not resolve the instability.

2 Likes

OMG! You made my day Yeah I can confirm that even starting from a kosher initial condition (e.g. very peaked Gaussian far from the boundaries) will result in instability UNLESS the magic sign turning is performed.

I thought it could be related to Dirac delta in boundary conditions for MethodOfLines.jl - #5 by xtalax but somehow even putting the time derivative on the other side does not help

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Great! It might still be worth opening an issue with MOL. @xtalax

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Thank you for your interest in MethodOfLines.jl .

This was a known issue in v0.3.0, but was fixed (I thought!) in v0.3.1. Which version of MOL are you using?

The first thing MOL does (in v0.3.1) is cast equations to a cardinal form:

pdeeqs = [eq.lhs - eq.rhs ~ 0 for eq in pdesys.eqs]


So this problem remaining would be very strange indeed.

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I have 0.3.1. As I said somehow even bringing everything on the lhs of the equation still makes it unstable.

Ah I see, there’s some weirdness going on here, Thank you for finding a valuable edge case!

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