[petsc-users] Best way to solve eqns for Runge Kutta 3rd order schemes

[TAY wee-beng]

Hi,

I need to solve the momentum eqns of my CFD code which uses the RK3 scheme.

I need to solve the x and y momentum eqns 3 times per time step.

Hence, for each sub time step, I have 2 linear eqns for x and y, which I 
solve using PETSc.

The off-diagonal elements of the matrix are different for each substep. 
But the off-diagonal elements for the same substep is the same for 
different time. ie

Off-diagonal elements (total 4) of the matrix of substep 1 at t=1 is the 
same as that of substep 1 at t=2,3....

Similarly, off-diagonal elements of the matrix of substep 2 at t=1 is 
the same as that of substep 2 at t=2,3....

Moreover, the off-diagonal elements of the matrix of substep 1,2 and 3 
differs by a fixed factor ie

if not considering the diagonal elements, matrix A(substep 1) == 
k1*A(substep 2) = k2*A(substep 3), where k1, k2 are constants

However, the diagonal elements of the matrix changes all the time, 
between substeps and between time steps.

The RHS vector also changes all the time.

I am not too sure what is the best way (most efficient, fastest) to 
solve these linear equations.

Currently, I write in the matrix at each substep. I doubt this is the 
best. I am thinking of:

1. Instead of creating 2 matrices for x and y, I create 6, 2 for each 
substep. The off-diagonal elements are always the same. I only use 
MatDiagonalSet to change the diagonal elements for each matrix at 
different time steps

or

2. I only create 2 matrices. At each substep, I multiply the matrices by 
a constant, so that off-diagonal elements at one substep is equal to 
that of another. Then I use MatDiagonalSet to do the same as above.

Which is a better mtd? Or is there other better mtds?

My main concern is to save time.

Moreover, how often do I have to do call:

call KSPCreate(MPI_COMM_WORLD,ksp_semi_x,ierr)

call KSPGetPC(ksp_semi_x,pc_semi_x,ierr)

ksptype=KSPBCGS

call KSPSetType(ksp_semi_x,ksptype,ierr)

call KSPSetFromOptions(ksp_semi_x,ierr)

tol=1.e-5

call 
KSPSetTolerances(ksp_semi_x,tol,PETSC_DEFAULT_DOUBLE_PRECISION,PETSC_DEFAULT_DOUBLE_PRECISION,PETSC_DEFAULT_INTEGER,ierr)

Thank you very much for the help!

-- 
Yours sincerely,

TAY wee-beng