# Help me outperform Matlab in numerical solution of semidiscretized PDE system as much as possible

**URL:** <https://discourse.julialang.org/t/help-me-outperform-matlab-in-numerical-solution-of-semidiscretized-pde-system-as-much-as-possible/76777>\
**Category:** Modelling & Simulations\
**Tags:** performance, pde, modelingtoolkit, differentialequation\
**Created:** [February 19, 2022, 10:05pm UTC](https://discourse.julialang.org/t/help-me-outperform-matlab-in-numerical-solution-of-semidiscretized-pde-system-as-much-as-possible/76777 "2022-02-19T22:05:44Z")\
**Posts on this page:** 1\
**Showing post:** 20

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**Author:** ![RobertS](https://avatars.discourse-cdn.com/v4/letter/r/ee7513/32.png) [@RobertS](https://discourse.julialang.org/u/RobertS)\
**Post date:** [February 28, 2022, 9:12pm UTC](https://discourse.julialang.org/t/help-me-outperform-matlab-in-numerical-solution-of-semidiscretized-pde-system-as-much-as-possible/76777/20 "2022-02-28T21:12:53Z")

</div>

Thank you for the assistance. I got the MTKized system to work

```julia

# build initial conditions
u0 = zeros(N*3 + 1)
u0[indices_Tgas] .= Tin
u0[indices_Tsol] .= 20
u0[indices_mflux] = 0.5
u0[indices_RConcentrationSolid] .= 3000

# build mass matrix
Ii = [indices_Tsol; indices_RConcentrationSolid];
Jj = [indices_Tsol; indices_RConcentrationSolid];
V = ones(length(Ii));
Msparse = sparse(Ii,Jj,V, 3*N+1, 3*N+1)

f_DAE = ODEFunction(DiscretizedSystemMassMatrixForm_noDualcache!, mass_matrix=Msparse)
probDAE = ODEProblem(f_DAE,u0,(0.0,1500),p_noDualCache)
de = modelingtoolkitize(probDAE)

# MTKized system back to numerical problem:
probDAE_MTK1 = ODEProblem(de,[],(0.0,1500),p_noDualCache,jac=true,sparse=true)

# solve the numerical problem - works
sol_MTK1 = solve(probDAE_MTK1,QBDF())

# pretty much as fast as solving non-MTKized system with Jacobian sparsity, as in above posts.
@benchmark solve(probDAE_MTK1,QBDF()) 

```

To my surpise, the MTKized system does not solve faster than the original system with Jacobian sparsity information, although the MTKized system should use the analytic Jacobian, or am I mistaken?

Anyway. In a next step, I tried to structual\_simplify the system and solve it:

```julia
# Now try to structural_simplify the system before solving it
de2 = structural_simplify(de)

probDAE_MTK2 = ODEProblem(de2,[],(0.0,1500),p_noDualCache,jac=true,sparse=true)

sol_MTK2 = solve(probDAE_MTK2,QBDF())

```

Unfortunately, I get an `UndefVarError: x201 not defined.` Why is that? What can I do to get it to work?

Thank you!

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