106
5. Solution of Linear Equation Systems
for the spatial finite differences. Rearranging Eq. (5.47), we find that, at time
step n + 1, we have to solve the system of equations:
As dn+' - 4" z At 8 4 / a t , the last term is proportional to (At)3 for small
At. Since the FD approximation is of second order, for small At, the last
term is small compared to the discretization error and may be neglected.
The remaining equation can be factored into two simpler equations:
Each of these systems of equations is a set of tridiagonal equations that can be
solved with the efficient TDMA method; this requires no iteration and is much
cheaper than solving Eq. (5.47). Either Eq. (5.49) or (5.50), as a method in its
own right, is only first-order accurate in time and conditionally stable but the
combined method is second-order accurate and unconditionally stable! The
family of methods based on these ideas are known as splitting or approximate
factorization methods; a wide variety of them has been developed.
Neglect of the third-order term, which is essential to the factorization,
is justified only when the time step is small. So, although the method is
unconditionally stable, it may not be accurate in time if the time step is large.
For elliptic equations, the objective is to obtain the steady state solution
as quickly as possible; this is best accomplished with the largest possible
time step. However, the factorization error becomes large when the time step
is large so the method loses some of its effectiveness. In fact, there is an
optimum time step which gives the most rapid convergence. When this time
step is used, the AD1 method is very efficient - it converges in a number of
iterations proportional to the number of points in one direction.
A better strategy uses different time steps for several iterations in a cyclic
fashion. This approach can make the number of iterations for convergence
proportional to the square root of the number of grid points in one direction,
making AD1 an excellent method.
Equations which involve the convection and source terms require some
generalization of this method. In CFD, the pressure or pressure-correction
5. Solution of Linear Equation Systems
for the spatial finite differences. Rearranging Eq. (5.47), we find that, at time
step n + 1, we have to solve the system of equations:
As dn+' - 4" z At 8 4 / a t , the last term is proportional to (At)3 for small
At. Since the FD approximation is of second order, for small At, the last
term is small compared to the discretization error and may be neglected.
The remaining equation can be factored into two simpler equations:
Each of these systems of equations is a set of tridiagonal equations that can be
solved with the efficient TDMA method; this requires no iteration and is much
cheaper than solving Eq. (5.47). Either Eq. (5.49) or (5.50), as a method in its
own right, is only first-order accurate in time and conditionally stable but the
combined method is second-order accurate and unconditionally stable! The
family of methods based on these ideas are known as splitting or approximate
factorization methods; a wide variety of them has been developed.
Neglect of the third-order term, which is essential to the factorization,
is justified only when the time step is small. So, although the method is
unconditionally stable, it may not be accurate in time if the time step is large.
For elliptic equations, the objective is to obtain the steady state solution
as quickly as possible; this is best accomplished with the largest possible
time step. However, the factorization error becomes large when the time step
is large so the method loses some of its effectiveness. In fact, there is an
optimum time step which gives the most rapid convergence. When this time
step is used, the AD1 method is very efficient - it converges in a number of
iterations proportional to the number of points in one direction.
A better strategy uses different time steps for several iterations in a cyclic
fashion. This approach can make the number of iterations for convergence
proportional to the square root of the number of grid points in one direction,
making AD1 an excellent method.
Equations which involve the convection and source terms require some
generalization of this method. In CFD, the pressure or pressure-correction