Solving Heat equation using Julia

New to Julia
,
1

Hi,
I am new user of Julia and I want to use it for solving PDEs and ODEs numerically. I am trying to run examples that are available in Julia website or GitHub but I get error.
For instance I want to run this example:

using OrdinaryDiffEq, ModelingToolkit, DiffEqOperators
# Method of Manufactured Solutions: exact solution
u_exact = (x,t) -> exp.(-t) * cos.(x)

# Parameters, variables, and derivatives
@parameters t x
@variables u(..)
Dt = Differential(t)
Dxx = Differential(x)^2

# 1D PDE and boundary conditions
eq  = Dt(u(t,x)) ~ Dxx(u(t,x))
bcs = [u(0,x) ~ cos(x),
        u(t,0) ~ exp(-t),
        u(t,1) ~ exp(-t) * cos(1)]

# Space and time domains
domains = [t ∈ IntervalDomain(0.0,1.0),
           x ∈ IntervalDomain(0.0,1.0)]

# PDE system
pdesys = PDESystem(eq,bcs,domains,[t,x],[u(t,x)])

# Method of lines discretization
dx = 0.1
order = 2
discretization = MOLFiniteDifference([x=>dx],t)

# Convert the PDE problem into an ODE problem
prob = discretize(pdesys,discretization)

# Solve ODE problem
using OrdinaryDiffEq
sol = solve(prob,Tsit5(),saveat=0.2)

# Plot results and compare with exact solution
x = (0:dx:1)[2:end-1]
t = sol.t

using Plots
plt = plot()

for i in 1:length(t)
    plot!(x,sol.u[i],label="Numerical, t=$(t[i])")
    scatter!(x, u_exact(x, t[i]),label="Exact, t=$(t[i])")
end
display(plt)
savefig("plot.png")

But I get this error:
UndefKeywordError: keyword argument name not assigned

Stacktrace:
[1] PDESystem(eqs::Equation, bcs::Vector{Equation}, domain::Vector{Symbolics.VarDomainPairing}, ivs::Vector{Num}, dvs::Vector{Num}, ps::SciMLBase.NullParameters) (repeats 2 times)
@ ModelingToolkit C:\Users\rm18124.julia\packages\ModelingToolkit\57XKa\src\systems\pde\pdesystem.jl:75
[2] top-level scope
@ In[32]:22
[3] eval
@ .\boot.jl:373 [inlined]
[4] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)
@ Base .\loading.jl:1196

I double checked the PDESystem and it looks fine, any help please?
Thanks

2 
@named pdesys = PDESystem(eq,bcs,domains,[t,x],[u(t,x)])

Any system needs to have a unique name. We can probably improve the error message there.

1 Like
4

Thank you, it worked well.

5

Here is the right code, in the above code is missing MethodOfLines package

using OrdinaryDiffEq, ModelingToolkit, DiffEqOperators, MethodOfLines
# Method of Manufactured Solutions: exact solution
u_exact = (x,t) -> exp.(-t) * cos.(x)

# Parameters, variables, and derivatives
@parameters t x
@variables u(..)
Dt = Differential(t)
Dxx = Differential(x)^2

# 1D PDE and boundary conditions
eq  = Dt(u(t,x)) ~ Dxx(u(t,x))
bcs = [u(0,x) ~ cos(x),
        u(t,0) ~ exp(-t),
        u(t,1) ~ exp(-t) * cos(1)]

# Space and time domains
domains = [t ∈ IntervalDomain(0.0,1.0),
           x ∈ IntervalDomain(0.0,1.0)]

# PDE system
@named pdesys  = PDESystem(eq,bcs,domains,[t,x],[u(t,x)])

# Method of lines discretization
dx = 0.1
order = 2
discretization = MOLFiniteDifference([x=>dx],t)

# Convert the PDE problem into an ODE problem
prob = discretize(pdesys,discretization)

# Solve ODE problem
using OrdinaryDiffEq
sol = solve(prob ,Tsit5(),saveat=0.2)

# Plot results and compare with exact solution
x = (0:dx:1)[2:end-1]
t = sol.t

using Plots
plt = plot()

for i in 1:length(t)
    plot!(x,sol.u[i],label="Numerical, t=$(t[i])")
    scatter!(x, u_exact(x, t[i]),label="Exact, t=$(t[i])")
end
display(plt)
savefig("plot.png")
6

@xtalax is this fixed in the docs?

7

Yes, caught this issue last week. Just waiting on merge, still getting random segfault in doctest though.