Re: [petsc-users] 'Preconditioning' with lower-order method

[Zou, Ling via petsc-users]

From: Zhang, Hong 
Date: Monday, March 4, 2024 at 6:34 PM
To: Zou, Ling 
Cc: Jed Brown , Barry Smith , 
petsc-users@mcs.anl.gov 
Subject: Re: [petsc-users] 'Preconditioning' with lower-order method
Ling,

Are you using PETSc TS? If so, it may worth trying Crank-Nicolson first to see 
if the nonlinear solve becomes faster.

>>>>> No, I’m not using TS, and I don’t plan to use CN. From my experience, 
>>>>> when dealing with (nearly) incompressible flow problems, CN often cause 
>>>>> (very large) pressure temporal oscillations, and to avoid that, the 
>>>>> pressure is often using fully implicit method, so that would cause quite 
>>>>> some code implementation issue.
For the pressure oscillation issue, also see page 7 of INL/EXT-12-27197. Notes 
on Newton-Krylov Based Incompressible Flow Projection Solver


In addition, you can try to improve the performance by pruning the Jacobian 
matrix.

TSPruneIJacobianColor()<https://urldefense.us/v3/__https://petsc.org/main/manualpages/TS/TSPruneIJacobianColor/__;!!G_uCfscf7eWS!aXVa0uz1LUIOdvZEPlRJOhRzz9h8MSM4vhl93kknxKGb8hkTyjCFJmSZIGr0fYx90rrqotBGdw-NwiF27Ec$
 > sometimes can reduce the number of colors especially for high-order methods 
and make your Jacobian matrix more compact. An example of usage can be found 
here<https://urldefense.us/v3/__https://petsc.org/release/src/ts/tests/ex5.c.html__;!!G_uCfscf7eWS!aXVa0uz1LUIOdvZEPlRJOhRzz9h8MSM4vhl93kknxKGb8hkTyjCFJmSZIGr0fYx90rrqotBGdw-NmisGfPQ$
 >. If you are not using TS, there is a SNES version 
SNESPruneJacobianColor()<https://urldefense.us/v3/__https://petsc.org/main/manualpages/SNES/SNESPruneJacobianColor/__;!!G_uCfscf7eWS!aXVa0uz1LUIOdvZEPlRJOhRzz9h8MSM4vhl93kknxKGb8hkTyjCFJmSZIGr0fYx90rrqotBGdw-NfYBUJqI$
 > for the same functionality.

>>>>> The following code is how I setup the coloring.
  {
// Create Matrix-free context
MatCreateSNESMF(snes, _MatrixFree);

// Let the problem setup Jacobian matrix sparsity
p_sim->FillJacobianMatrixNonZeroEntry(P_Mat);

// See PETSc examples:
// 
https://urldefense.us/v3/__https://petsc.org/release/src/snes/tutorials/ex14.c.html__;!!G_uCfscf7eWS!aXVa0uz1LUIOdvZEPlRJOhRzz9h8MSM4vhl93kknxKGb8hkTyjCFJmSZIGr0fYx90rrqotBGdw-N3ZHE6Qw$
 
// 
https://urldefense.us/v3/__https://petsc.org/release/src/mat/tutorials/ex16.c.html__;!!G_uCfscf7eWS!aXVa0uz1LUIOdvZEPlRJOhRzz9h8MSM4vhl93kknxKGb8hkTyjCFJmSZIGr0fYx90rrqotBGdw-NcpSrhMM$
 
ISColoring iscoloring;
MatColoring mc;
MatColoringCreate(P_Mat, );
MatColoringSetType(mc, MATCOLORINGSL);
MatColoringSetFromOptions(mc);
MatColoringApply(mc, );
MatColoringDestroy();
MatFDColoringCreate(P_Mat, iscoloring, );
MatFDColoringSetFunction(
fdcoloring, (PetscErrorCode(*)(void))(void (*)(void))snesFormFunction, 
this);
MatFDColoringSetFromOptions(fdcoloring);
MatFDColoringSetUp(P_Mat, iscoloring, fdcoloring);
ISColoringDestroy();

// Should I prune here? Like

SNESPruneJacobianColor<https://urldefense.us/v3/__https://petsc.org/main/manualpages/SNES/SNESPruneJacobianColor/__;!!G_uCfscf7eWS!aXVa0uz1LUIOdvZEPlRJOhRzz9h8MSM4vhl93kknxKGb8hkTyjCFJmSZIGr0fYx90rrqotBGdw-NfYBUJqI$
 >(snes, P_Mat, P_Mat);


SNESSetJacobian(snes,// snes
J_MatrixFree,// Jacobian-free
P_Mat,   // Preconditioning matrix
SNESComputeJacobianDefaultColor, // Use finite differencing 
and coloring
fdcoloring); // fdcoloring
  }

Thanks,

-Ling




Hong (Mr.)


On Mar 3, 2024, at 11:48 PM, Zou, Ling via petsc-users 
 wrote:



From: Jed Brown mailto:j...@jedbrown.org>>
Date: Sunday, March 3, 2024 at 11:35 PM
To: Zou, Ling mailto:l...@anl.gov>>, Barry Smith 
mailto:bsm...@petsc.dev>>
Cc: petsc-users@mcs.anl.gov<mailto:petsc-users@mcs.anl.gov> 
mailto:petsc-users@mcs.anl.gov>>
Subject: Re: [petsc-users] 'Preconditioning' with lower-order method
If you're having PETSc use coloring and have confirmed that the stencil is 
sufficient, then it would be nonsmoothness (again, consider the limiter you've 
chosen) preventing quadratic convergence (assuming that doesn't kick in 
eventually). Note
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If you're having PETSc use coloring and have confirmed that the stencil is 
sufficient, then it would be nonsmoothness (again, consider the limiter you've 
chosen) preventing quadratic convergence (assuming that doesn't kick in 
eventually).



• Yes, I do use coloring, and I do provide sufficient stencil, i.e., 
neighbor’s neighbor. The sufficiency is confirmed by PETSc’s 
-snes_test_jacobian and -snes_test_jacobian_view options.



Note that assem

Re: [petsc-users] 'Preconditioning' with lower-order method

[Zhang, Hong via petsc-users]

Ling,

Are you using PETSc TS? If so, it may worth trying Crank-Nicolson first to see 
if the nonlinear solve becomes faster.

In addition, you can try to improve the performance by pruning the Jacobian 
matrix.

TSPruneIJacobianColor()<https://urldefense.us/v3/__https://petsc.org/main/manualpages/TS/TSPruneIJacobianColor/__;!!G_uCfscf7eWS!aAgMOjDD6ZgyEaxqc_vH-Yl5P1uNvFWa8_Lc5TWgoXfSz9x7kNgCi4jr10b2PmgzFwIyeCZGCGLUYU6hNJdYW8zcNQ$
 > sometimes can reduce the number of colors especially for high-order methods 
and make your Jacobian matrix more compact. An example of usage can be found 
here<https://urldefense.us/v3/__https://petsc.org/release/src/ts/tests/ex5.c.html__;!!G_uCfscf7eWS!aAgMOjDD6ZgyEaxqc_vH-Yl5P1uNvFWa8_Lc5TWgoXfSz9x7kNgCi4jr10b2PmgzFwIyeCZGCGLUYU6hNJd4iC6hZw$
 >. If you are not using TS, there is a SNES version 
SNESPruneJacobianColor()<https://urldefense.us/v3/__https://petsc.org/main/manualpages/SNES/SNESPruneJacobianColor/__;!!G_uCfscf7eWS!aAgMOjDD6ZgyEaxqc_vH-Yl5P1uNvFWa8_Lc5TWgoXfSz9x7kNgCi4jr10b2PmgzFwIyeCZGCGLUYU6hNJfq48Io5Q$
 > for the same functionality.

Hong (Mr.)

On Mar 3, 2024, at 11:48 PM, Zou, Ling via petsc-users 
 wrote:



From: Jed Brown mailto:j...@jedbrown.org>>
Date: Sunday, March 3, 2024 at 11:35 PM
To: Zou, Ling mailto:l...@anl.gov>>, Barry Smith 
mailto:bsm...@petsc.dev>>
Cc: petsc-users@mcs.anl.gov<mailto:petsc-users@mcs.anl.gov> 
mailto:petsc-users@mcs.anl.gov>>
Subject: Re: [petsc-users] 'Preconditioning' with lower-order method
If you're having PETSc use coloring and have confirmed that the stencil is 
sufficient, then it would be nonsmoothness (again, consider the limiter you've 
chosen) preventing quadratic convergence (assuming that doesn't kick in 
eventually). Note
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If you're having PETSc use coloring and have confirmed that the stencil is 
sufficient, then it would be nonsmoothness (again, consider the limiter you've 
chosen) preventing quadratic convergence (assuming that doesn't kick in 
eventually).



• Yes, I do use coloring, and I do provide sufficient stencil, i.e., 
neighbor’s neighbor. The sufficiency is confirmed by PETSc’s 
-snes_test_jacobian and -snes_test_jacobian_view options.



Note that assembling a Jacobian of a second order TVD operator requires at 
least second neighbors while the first order needs only first neighbors, thus 
is much sparser and needs fewer colors to compute.



• In my code implementation, when marking the Jacobian nonzero pattern, 
I don’t differentiate FV1 or FV2, I always use the FV2 stencil, so it’s a bit 
‘fat’ for the FV1 method, but worked just fine.



I expect you're either not exploiting that in the timings or something else is 
amiss. You can run with `-log_view -snes_view -ksp_converged_reason` to get a 
bit more information about what's happening.



• The attached is screen output as you suggest. The linear and 
nonlinear performance of FV2 is both worse from the output.



FV2:

Time Step 149, time = 13229.7, dt = 100
NL Step =  0, fnorm =  7.80968E-03
  Linear solve converged due to CONVERGED_RTOL iterations 26
NL Step =  1, fnorm =  7.65731E-03
  Linear solve converged due to CONVERGED_RTOL iterations 24
NL Step =  2, fnorm =  6.85034E-03
  Linear solve converged due to CONVERGED_RTOL iterations 27
NL Step =  3, fnorm =  6.11873E-03
  Linear solve converged due to CONVERGED_RTOL iterations 25
NL Step =  4, fnorm =  1.57347E-03
  Linear solve converged due to CONVERGED_RTOL iterations 27
NL Step =  5, fnorm =  9.03536E-04
SNES Object: 1 MPI process
  type: newtonls
  maximum iterations=20, maximum function evaluations=1
  tolerances: relative=1e-08, absolute=1e-06, solution=1e-08
  total number of linear solver iterations=129
  total number of function evaluations=144
  norm schedule ALWAYS
  Jacobian is applied matrix-free with differencing
  Preconditioning Jacobian is built using finite differences with coloring
  SNESLineSearch Object: 1 MPI process
type: bt
  interpolation: cubic
  alpha=1.00e-04
maxstep=1.00e+08, minlambda=1.00e-12
tolerances: relative=1.00e-08, absolute=1.00e-15, 
lambda=1.00e-08
maximum iterations=40
  KSP Object: 1 MPI process
type: gmres
  restart=100, using Classical (unmodified) Gram-Schmidt Orthogonalization 
with no iterative refinement
  happy breakdown tolerance 1e-30
maximum iterations=100, initial guess is zero
tolerances:  relative=0.0001, absolute=1e-50, divergence=1.
left preconditioning
using PRECONDITIONED norm type for convergence test
  PC Object: 1 MPI process
type: ilu
  out-of-place factorization
  0 levels of fill
  tolerance for zero pivot 2.22045e-14
  using diagonal shift to prevent zero pivot [NONZERO]
  matrix orderi

Re: [petsc-users] 'Preconditioning' with lower-order method

[Zou, Ling via petsc-users]

From: Jed Brown 
Date: Sunday, March 3, 2024 at 11:35 PM
To: Zou, Ling , Barry Smith 
Cc: petsc-users@mcs.anl.gov 
Subject: Re: [petsc-users] 'Preconditioning' with lower-order method
If you're having PETSc use coloring and have confirmed that the stencil is 
sufficient, then it would be nonsmoothness (again, consider the limiter you've 
chosen) preventing quadratic convergence (assuming that doesn't kick in 
eventually). Note
ZjQcmQRYFpfptBannerStart
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If you're having PETSc use coloring and have confirmed that the stencil is 
sufficient, then it would be nonsmoothness (again, consider the limiter you've 
chosen) preventing quadratic convergence (assuming that doesn't kick in 
eventually).



· Yes, I do use coloring, and I do provide sufficient stencil, i.e., 
neighbor’s neighbor. The sufficiency is confirmed by PETSc’s 
-snes_test_jacobian and -snes_test_jacobian_view options.



Note that assembling a Jacobian of a second order TVD operator requires at 
least second neighbors while the first order needs only first neighbors, thus 
is much sparser and needs fewer colors to compute.



· In my code implementation, when marking the Jacobian nonzero pattern, 
I don’t differentiate FV1 or FV2, I always use the FV2 stencil, so it’s a bit 
‘fat’ for the FV1 method, but worked just fine.



I expect you're either not exploiting that in the timings or something else is 
amiss. You can run with `-log_view -snes_view -ksp_converged_reason` to get a 
bit more information about what's happening.



· The attached is screen output as you suggest. The linear and 
nonlinear performance of FV2 is both worse from the output.



FV2:

Time Step 149, time = 13229.7, dt = 100

NL Step =  0, fnorm =  7.80968E-03

  Linear solve converged due to CONVERGED_RTOL iterations 26

NL Step =  1, fnorm =  7.65731E-03

  Linear solve converged due to CONVERGED_RTOL iterations 24

NL Step =  2, fnorm =  6.85034E-03

  Linear solve converged due to CONVERGED_RTOL iterations 27

NL Step =  3, fnorm =  6.11873E-03

  Linear solve converged due to CONVERGED_RTOL iterations 25

NL Step =  4, fnorm =  1.57347E-03

  Linear solve converged due to CONVERGED_RTOL iterations 27

NL Step =  5, fnorm =  9.03536E-04

SNES Object: 1 MPI process

  type: newtonls

  maximum iterations=20, maximum function evaluations=1

  tolerances: relative=1e-08, absolute=1e-06, solution=1e-08

  total number of linear solver iterations=129

  total number of function evaluations=144

  norm schedule ALWAYS

  Jacobian is applied matrix-free with differencing

  Preconditioning Jacobian is built using finite differences with coloring

  SNESLineSearch Object: 1 MPI process

type: bt

  interpolation: cubic

  alpha=1.00e-04

maxstep=1.00e+08, minlambda=1.00e-12

tolerances: relative=1.00e-08, absolute=1.00e-15, 
lambda=1.00e-08

maximum iterations=40

  KSP Object: 1 MPI process

type: gmres

  restart=100, using Classical (unmodified) Gram-Schmidt Orthogonalization 
with no iterative refinement

  happy breakdown tolerance 1e-30

maximum iterations=100, initial guess is zero

tolerances:  relative=0.0001, absolute=1e-50, divergence=1.

left preconditioning

using PRECONDITIONED norm type for convergence test

  PC Object: 1 MPI process

type: ilu

  out-of-place factorization

  0 levels of fill

  tolerance for zero pivot 2.22045e-14

  using diagonal shift to prevent zero pivot [NONZERO]

  matrix ordering: rcm

  factor fill ratio given 1., needed 1.

Factored matrix follows:

  Mat Object: 1 MPI process

type: seqaij

rows=8715, cols=8715

package used to perform factorization: petsc

total: nonzeros=38485, allocated nonzeros=38485

  not using I-node routines

linear system matrix followed by preconditioner matrix:

Mat Object: 1 MPI process

  type: mffd

  rows=8715, cols=8715

Matrix-free approximation:

  err=1.49012e-08 (relative error in function evaluation)

  Using wp compute h routine

  Does not compute normU

Mat Object: 1 MPI process

  type: seqaij

  rows=8715, cols=8715

  total: nonzeros=38485, allocated nonzeros=38485

  total number of mallocs used during MatSetValues calls=0

not using I-node routines

 Solve Converged!





FV1:

Time Step 149, time = 13229.7, dt = 100

NL Step =  0, fnorm =  7.90072E-03

  Linear solve converged due to CONVERGED_RTOL iterations 12

NL Step =  1, fnorm =  2.01919E-04

  Linear solve converged due to CONVERGED_RTOL iterations 17

NL Step =  2, fnorm =  1.06960E-05

  Linear solve converged due to CONVERGED_RTOL iterations 15

NL Step =  3, fnorm =  2.41683E-09

SNES

Re: [petsc-users] 'Preconditioning' with lower-order method

[Jed Brown]

 If you're having PETSc use coloring and have confirmed that the stencil is sufficient, then it would be nonsmoothness (again, consider the limiter you've chosen) preventing quadratic convergence (assuming that doesn't kick in eventually). Note




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If you're having PETSc use coloring and have confirmed that the stencil is sufficient, then it would be nonsmoothness (again, consider the limiter you've chosen) preventing quadratic convergence (assuming that doesn't kick in eventually). Note that assembling a Jacobian of a second order TVD operator requires at least second neighbors while the first order needs only first neighbors, thus is much sparser and needs fewer colors to compute. I expect you're either not exploiting that in the timings or something else is amiss. You can run with `-log_view -snes_view -ksp_converged_reason` to get a bit more information about what's happening. 

"Zou, Ling via petsc-users"  writes:

> Barry, thank you.
> I am not sure if I exactly follow you on this:
> “Are you forming the Jacobian for the first and second order cases inside of Newton?”
>
> The problem that we deal with, heat/mass transfer in heterogeneous systems (reactor system), is generally small in terms of size, i.e., # of DOFs (several k to maybe 100k level), so for now, I completely rely on PETSc to compute Jacobian, i.e., finite-differencing.
>
> That’s a good suggestion to see the time spent during various events.
> What motivated me to try the options are the following observations.
>
> 2nd order FVM:
>
> Time Step 149, time = 13229.7, dt = 100
>
> NL Step =  0, fnorm =  7.80968E-03
>
> NL Step =  1, fnorm =  7.65731E-03
>
> NL Step =  2, fnorm =  6.85034E-03
>
> NL Step =  3, fnorm =  6.11873E-03
>
> NL Step =  4, fnorm =  1.57347E-03
>
> NL Step =  5, fnorm =  9.03536E-04
>
>  Solve Converged!
>
> 1st order FVM:
>
> Time Step 149, time = 13229.7, dt = 100
>
> NL Step =  0, fnorm =  7.90072E-03
>
> NL Step =  1, fnorm =  2.01919E-04
>
> NL Step =  2, fnorm =  1.06960E-05
>
> NL Step =  3, fnorm =  2.41683E-09
>
>  Solve Converged!
>
> Notice the obvious ‘stagnant’ in residual for the 2nd order method while not in the 1st order.
> For the same problem, the wall time is 10 sec vs 6 sec. I would be happy if I can reduce 2 sec for the 2nd order method.
>
> -Ling
>
> From: Barry Smith 
> Date: Sunday, March 3, 2024 at 12:06 PM
> To: Zou, Ling 
> Cc: petsc-users@mcs.anl.gov 
> Subject: Re: [petsc-users] 'Preconditioning' with lower-order method
> Are you forming the Jacobian for the first and second order cases inside of Newton? You can run both with -log_view to see how much time is spent in the various events (compute function, compute Jacobian, linear solve, .. . ) for the two cases
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>
>Are you forming the Jacobian for the first and second order cases inside of Newton?
>
>You can run both with -log_view to see how much time is spent in the various events (compute function, compute Jacobian, linear solve, ...) for the two cases and compare them.
>
>
>
>
> On Mar 3, 2024, at 11:42 AM, Zou, Ling via petsc-users  wrote:
>
> Original email may have been sent to the incorrect place.
> See below.
>
> -Ling
>
> From: Zou, Ling >
> Date: Sunday, March 3, 2024 at 10:34 AM
> To: petsc-users >
> Subject: 'Preconditioning' with lower-order method
> Hi all,
>
> I am solving a PDE system over a spatial domain. Numerical methods are:
>
>   *   Finite volume method (both 1st and 2nd order implemented)
>   *   BDF1 and BDF2 for time integration.
> What I have noticed is that 1st order FVM converges much faster than 2nd order FVM, regardless the time integration scheme. Well, not surprising since 2nd order FVM introduces additional non-linearity.
>
> I’m thinking about two possible ways to speed up 2nd order FVM, and would like to get some thoughts or community knowledge before jumping into code implementation.
>
> Say, let the 2nd order FVM residual function be F2(x) = 0; and the 1st order FVM residual function be F1(x) = 0.
>
>   1.  Option – 1, multi-step for each time step
> Step 1: solving F1(x) = 0 to obtain a temporary solution x1
> Step 2: feed x1 as an initial guess to solve F2(x) = 0 to obtain the final solution.
> [Not sure if gain any saving at all]
>
>
>   1.  Option -2, dynamically changing residual function F(x)
> In ps

Re: [petsc-users] 'Preconditioning' with lower-order method

[Zou, Ling via petsc-users]

Barry, thank you.
I am not sure if I exactly follow you on this:
“Are you forming the Jacobian for the first and second order cases inside of 
Newton?”

The problem that we deal with, heat/mass transfer in heterogeneous systems 
(reactor system), is generally small in terms of size, i.e., # of DOFs (several 
k to maybe 100k level), so for now, I completely rely on PETSc to compute 
Jacobian, i.e., finite-differencing.

That’s a good suggestion to see the time spent during various events.
What motivated me to try the options are the following observations.

2nd order FVM:

Time Step 149, time = 13229.7, dt = 100

NL Step =  0, fnorm =  7.80968E-03

NL Step =  1, fnorm =  7.65731E-03

NL Step =  2, fnorm =  6.85034E-03

NL Step =  3, fnorm =  6.11873E-03

NL Step =  4, fnorm =  1.57347E-03

NL Step =  5, fnorm =  9.03536E-04

 Solve Converged!

1st order FVM:

Time Step 149, time = 13229.7, dt = 100

NL Step =  0, fnorm =  7.90072E-03

NL Step =  1, fnorm =  2.01919E-04

NL Step =  2, fnorm =  1.06960E-05

NL Step =  3, fnorm =  2.41683E-09

 Solve Converged!

Notice the obvious ‘stagnant’ in residual for the 2nd order method while not in 
the 1st order.
For the same problem, the wall time is 10 sec vs 6 sec. I would be happy if I 
can reduce 2 sec for the 2nd order method.

-Ling

From: Barry Smith 
Date: Sunday, March 3, 2024 at 12:06 PM
To: Zou, Ling 
Cc: petsc-users@mcs.anl.gov 
Subject: Re: [petsc-users] 'Preconditioning' with lower-order method
Are you forming the Jacobian for the first and second order cases inside of 
Newton? You can run both with -log_view to see how much time is spent in the 
various events (compute function, compute Jacobian, linear solve, .. . ) for 
the two cases
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   Are you forming the Jacobian for the first and second order cases inside of 
Newton?

   You can run both with -log_view to see how much time is spent in the various 
events (compute function, compute Jacobian, linear solve, ...) for the two 
cases and compare them.




On Mar 3, 2024, at 11:42 AM, Zou, Ling via petsc-users 
 wrote:

Original email may have been sent to the incorrect place.
See below.

-Ling

From: Zou, Ling mailto:l...@anl.gov>>
Date: Sunday, March 3, 2024 at 10:34 AM
To: petsc-users 
mailto:petsc-users-boun...@mcs.anl.gov>>
Subject: 'Preconditioning' with lower-order method
Hi all,

I am solving a PDE system over a spatial domain. Numerical methods are:

  *   Finite volume method (both 1st and 2nd order implemented)
  *   BDF1 and BDF2 for time integration.
What I have noticed is that 1st order FVM converges much faster than 2nd order 
FVM, regardless the time integration scheme. Well, not surprising since 2nd 
order FVM introduces additional non-linearity.

I’m thinking about two possible ways to speed up 2nd order FVM, and would like 
to get some thoughts or community knowledge before jumping into code 
implementation.

Say, let the 2nd order FVM residual function be F2(x) = 0; and the 1st order 
FVM residual function be F1(x) = 0.

  1.  Option – 1, multi-step for each time step
Step 1: solving F1(x) = 0 to obtain a temporary solution x1
Step 2: feed x1 as an initial guess to solve F2(x) = 0 to obtain the final 
solution.
[Not sure if gain any saving at all]


  1.  Option -2, dynamically changing residual function F(x)
In pseudo code, would be something like.

snesFormFunction(SNES snes, Vec u, Vec f, void *)
{
  if (snes.nl_it_no < 4) // 4 being arbitrary here
f = F1(u);
  else
f = F2(u);
}

I know this might be a bit crazy since it may crash after switching residual 
function, still, any thoughts?

Best,

-Ling

Re: [petsc-users] 'Preconditioning' with lower-order method

[Barry Smith]

   Are you forming the Jacobian for the first and second order cases inside of 
Newton?

   You can run both with -log_view to see how much time is spent in the various 
events (compute function, compute Jacobian, linear solve, ...) for the two 
cases and compare them.



> On Mar 3, 2024, at 11:42 AM, Zou, Ling via petsc-users 
>  wrote:
> 
> Original email may have been sent to the incorrect place.
> See below.
>  
> -Ling
>  
> From: Zou, Ling mailto:l...@anl.gov>>
> Date: Sunday, March 3, 2024 at 10:34 AM
> To: petsc-users  >
> Subject: 'Preconditioning' with lower-order method
> 
> Hi all,
>  
> I am solving a PDE system over a spatial domain. Numerical methods are:
> Finite volume method (both 1st and 2nd order implemented)
> BDF1 and BDF2 for time integration.
> What I have noticed is that 1st order FVM converges much faster than 2nd 
> order FVM, regardless the time integration scheme. Well, not surprising since 
> 2nd order FVM introduces additional non-linearity.
>  
> I’m thinking about two possible ways to speed up 2nd order FVM, and would 
> like to get some thoughts or community knowledge before jumping into code 
> implementation.
>  
> Say, let the 2nd order FVM residual function be F2(x) = 0; and the 1st order 
> FVM residual function be F1(x) = 0.
> Option – 1, multi-step for each time step
> Step 1: solving F1(x) = 0 to obtain a temporary solution x1
> Step 2: feed x1 as an initial guess to solve F2(x) = 0 to obtain the final 
> solution.
> [Not sure if gain any saving at all]
>  
> Option -2, dynamically changing residual function F(x)
> In pseudo code, would be something like.
>  
> snesFormFunction(SNES snes, Vec u, Vec f, void *)
> {
>   if (snes.nl_it_no < 4) // 4 being arbitrary here
> f = F1(u);
>   else
> f = F2(u);
> }
>  
> I know this might be a bit crazy since it may crash after switching residual 
> function, still, any thoughts?
>  
> Best,
>  
> -Ling