130
5. Solution of Linear Equation Systems
SIP is about four times as fast as LGS-ADI. For the GS and LGS solvers, the
number of iterations increases by about a factor of four each time the grid
is refined; the factor is smaller in case of SIP and LGS-ADI, but, as we shall
see in the next example, the factor increases and asymptotically approaches
four in the limit of very fine grids.
Another interesting observation is that the GS and LGS solvers do not
converge on a 20 x 20 CV grid with CDS discretization. This is due to the fact
that the matrix is not diagonally dominant in this case. Even on the 40 x 40
CV grid, the matrix is not completely diagonally dominant, but the violation
is in the region of uniform distribution of the variable (low gradients) so the
effect on the solver is not as severe. The LGS-AD1 and SIP solvers are not
affected.
We turn now to a test case for which an analytical solution exists and
the CDS approximation produces exact solution on any grid. This helps in
the evaluation of the iteration error but the behavior of the solver does not
benefit so this case is quite suitable for assessing solver performance. We are
solving the Laplace equation with Dirichlet boundary conditions, for which
the exact solution is 4 = xy. The solution domain is a rectangle, the solution
is prescribed at all boundaries and the initial values in the interior are all
zero. The initial error is thus equal to the solution and is a smooth function of
spatial coordinates. Discretization is performed using FV method described
in the previous section and CDS scheme. Since the convection is absent, the
problem is fully elliptic.
The solvers considered are:
Gauss-Seidel solver, denoted by GS;
Line Gauss-Seidel solver using TDMA alternately along lines x = const.
and y = const., denoted by LGS-ADI;
AD1 solver described in Sect. 5.3.5;
Stone's ILU method, denoted by SIP;
Conjugate gradient method, preconditioned using incomplete Cholesky decomposition, denoted ICCG;
Multigrid method using GS as a smoother, denoted MG-GS;
Multigrid method using SIP as a smoother, denoted MG-SIP.
Table 5.2 shows results obtained on uniform grids on a square solution
domain. LGS-AD1 is again about four times as fast as GS, and the SIP is
about four times as fast as LGS-ADI. AD1 is less efficient than SIP on coarse
grids but, when the optimum time step is chosen, the number of iterations
only doubles when the number of grid points in one direction is doubled, so
for fine grids it is quite effective. When the time step is varied in a cyclic
fashion, the solver becomes even more efficient. This is also true of SIP, but
cyclic variation of the parameter increases the cost per iteration. The number
of iterations required by AD1 to reach convergence for various time steps is
given in Table 5.3. The optimum time step is reduced by a factor of two when
the grid is refined.
5. Solution of Linear Equation Systems
SIP is about four times as fast as LGS-ADI. For the GS and LGS solvers, the
number of iterations increases by about a factor of four each time the grid
is refined; the factor is smaller in case of SIP and LGS-ADI, but, as we shall
see in the next example, the factor increases and asymptotically approaches
four in the limit of very fine grids.
Another interesting observation is that the GS and LGS solvers do not
converge on a 20 x 20 CV grid with CDS discretization. This is due to the fact
that the matrix is not diagonally dominant in this case. Even on the 40 x 40
CV grid, the matrix is not completely diagonally dominant, but the violation
is in the region of uniform distribution of the variable (low gradients) so the
effect on the solver is not as severe. The LGS-AD1 and SIP solvers are not
affected.
We turn now to a test case for which an analytical solution exists and
the CDS approximation produces exact solution on any grid. This helps in
the evaluation of the iteration error but the behavior of the solver does not
benefit so this case is quite suitable for assessing solver performance. We are
solving the Laplace equation with Dirichlet boundary conditions, for which
the exact solution is 4 = xy. The solution domain is a rectangle, the solution
is prescribed at all boundaries and the initial values in the interior are all
zero. The initial error is thus equal to the solution and is a smooth function of
spatial coordinates. Discretization is performed using FV method described
in the previous section and CDS scheme. Since the convection is absent, the
problem is fully elliptic.
The solvers considered are:
Gauss-Seidel solver, denoted by GS;
Line Gauss-Seidel solver using TDMA alternately along lines x = const.
and y = const., denoted by LGS-ADI;
AD1 solver described in Sect. 5.3.5;
Stone's ILU method, denoted by SIP;
Conjugate gradient method, preconditioned using incomplete Cholesky decomposition, denoted ICCG;
Multigrid method using GS as a smoother, denoted MG-GS;
Multigrid method using SIP as a smoother, denoted MG-SIP.
Table 5.2 shows results obtained on uniform grids on a square solution
domain. LGS-AD1 is again about four times as fast as GS, and the SIP is
about four times as fast as LGS-ADI. AD1 is less efficient than SIP on coarse
grids but, when the optimum time step is chosen, the number of iterations
only doubles when the number of grid points in one direction is doubled, so
for fine grids it is quite effective. When the time step is varied in a cyclic
fashion, the solver becomes even more efficient. This is also true of SIP, but
cyclic variation of the parameter increases the cost per iteration. The number
of iterations required by AD1 to reach convergence for various time steps is
given in Table 5.3. The optimum time step is reduced by a factor of two when
the grid is refined.