Hello,

I have been looking at the notebook https://benchmarks.sciml.ai/html/MOLPDE/ks_fdm_wpd.html that solves the Kuramot-Sivashinsky (KS) problem with finite differences.

The 1D equation is:

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial x^2} + \frac{\partial^4 u}{\partial x^4} = 0.

I am not sure about certain coefficients in `lin_term`

:

```
DiffEqArrayOperator(-0.0004*((1/dx^2) * diagm(-1 => du2, 0 => d2, 1 => du2)
+(1/dx^4) * diagm(-2 => duu4, -1 => du4, 0 => d4, 1 => du4, 2 => duu4)))
```

Why is there a `-0.0004`

coefficient?

Also, I would expect to have a minus sign in '(1/dx^4) * diagm(-2 => duu4, -1 => du4, 0 => d4, 1 => du4, 2 => duu4))``` since this term got moves to the right hand side .

The full code from https://benchmarks.sciml.ai/html/MOLPDE/ks_fdm_wpd.html

```
using ApproxFun, OrdinaryDiffEq, Sundials
using DiffEqDevTools
using LinearAlgebra
using Plots; gr()
# Define the linear and nonlinear terms
function lin_term(N)
#which is -(D2+D4)
dx = 1/(N + 1)
d2 = (-2) * ones(N) # main diagonal
du2 = ones(N - 1) # off diagonal
d4 = 6 * ones(N) # main diagonal
du4 = (-4) * ones(N - 1) # off diagonal
duu4 = ones(N - 2)
DiffEqArrayOperator(-0.0004*((1/dx^2) * diagm(-1 => du2, 0 => d2, 1 => du2)
+(1/dx^4) * diagm(-2 => duu4, -1 => du4, 0 => d4, 1 => du4, 2 => duu4)))
end
function nl_term(N)
dx = 1/(N + 1)
du = ones(N - 1) # super diagonal
dl = -ones(N - 1) # lower diagonal
D = (-0.2/(4*dx)) * diagm(-1 => dl, 1 => du)
tmp = zeros(N)
function (du,u,p,t)
@. tmp = u^2
mul!(du, D, tmp)
end
end
# Construct the problem
function ks(N)
f1 = lin_term(N)
f2 = nl_term(N)
dx = 1 / (N + 1)
xs = (1:N) * dx
μ0 = 0.3; σ0 = 0.05
f0 = x -> 0.6*exp(-(x - μ0)^2 / (2 * σ0^2))
u0 = f0.(xs)
prob = SplitODEProblem(f1, f2, u0, (0.0, 1.0))
xs, prob
end;
xs, prob = ks(100)
sol = solve(prob, RadauIIA5(autodiff=false); abstol=1e-14, reltol=1e-14)
test_sol = TestSolution(sol);
tslices = [0.0 0.25 0.50 0.75 1.]
ys = hcat((sol(t) for t in tslices)...)
labels = ["t = $t" for t in tslices]
plot(xs, ys, label=labels)
```