Issue with ode solver for solving burgers equation using spectral method

using DifferentialEquations
using LinearAlgebra

function burgers!(du, u, p, t)
    N = length(u)
    du[1] = 0.0
    du[N] = 0.0
    for i in 2:N-1
        du[i] = -u[i]*(u[i+1] - u[i-1])/2
    end
end

N = 100
x = range(0, stop=2*pi, length=N)
u0 = sin.(x)
tspan = (0.0, 1.0)
prob = ODEProblem(burgers!, u0, tspan)
sol = solve(prob, Tsit5())

using Plots
plot(sol, xaxis="x", yaxis="u(x)", label="Solution")

Option 1: Increase the number of grid points

One possible solution to the issue with the ode solver for solving the burgers equation using the spectral method is to increase the number of grid points. By increasing the number of grid points, we can obtain a more accurate approximation of the solution.

N = 200
x = range(0, stop=2*pi, length=N)
u0 = sin.(x)
tspan = (0.0, 1.0)
prob = ODEProblem(burgers!, u0, tspan)
sol = solve(prob, Tsit5())

using Plots
plot(sol, xaxis="x", yaxis="u(x)", label="Solution")

Option 2: Change the solver algorithm

Another possible solution is to change the solver algorithm. In this case, we can try using the `CVODE_BDF` solver algorithm instead of the default `Tsit5` algorithm.

N = 100
x = range(0, stop=2*pi, length=N)
u0 = sin.(x)
tspan = (0.0, 1.0)
prob = ODEProblem(burgers!, u0, tspan)
sol = solve(prob, CVODE_BDF())

using Plots
plot(sol, xaxis="x", yaxis="u(x)", label="Solution")

Option 3: Use a different spectral method

Lastly, we can try using a different spectral method to solve the burgers equation. Instead of using the default spectral method, we can try using a Chebyshev spectral method.

using ApproxFun

N = 100
x = Fun(identity, Chebyshev(0..2*pi))
u0 = Fun(sin, Chebyshev(0..2*pi))
tspan = (0.0, 1.0)
prob = ODEProblem(burgers!, u0, tspan)
sol = solve(prob, Tsit5())

using Plots
plot(sol, xaxis="x", yaxis="u(x)", label="Solution")

Among the three options, the best solution depends on the specific requirements and constraints of the problem. If accuracy is the main concern, increasing the number of grid points (Option 1) would be the best choice. If efficiency is more important, changing the solver algorithm (Option 2) might be a better option. Lastly, if the spectral method itself is causing issues, using a different spectral method (Option 3) can be considered. It is recommended to try out different options and evaluate their performance based on the specific problem at hand.