Time-Splitting Methods

Another approach developed for solving the advective-diffusion equation is the fractional step

technique, also known as time-splitting or operator splitting technique. This technique transforms the solution of a complicated time dependent partial differential equation into the sequential solution of simpler problems. For example, consider the following form of the one-dimensional advection-diffusion equation

dc

~di Kc ± S

to be solved numerically over one time step. This may be written as the set of equations (see, for example Belleudy and Sauvaget [1985])

1 dc dc ---+ u — 3 dt dx = 0, 1 dc 3 ~dt = 0 and 1 dc „ --- + Kc 3 dt 5 = 0.

The advection process is concentrated in one equation, with the diffusion process in another and the reaction and source-sink terms in an ordinary differential equation. The sum of these equations yield the original equation being modelled.

The split operator approach, in which the advection and diffusion are computed independently, has been pursued because of its advantages in allowing the use of accurate methods for each process. In particular, this has made it possible to exploit the hyperbolic nature of advection to devise characteristic based schemes (see, for example Belleudy and Sauvaget [1985] and Yang and Hsu [1991])

In practical problems, the reaction process is nonlinear and occurs more rapidly than either the advection and diffusion process. To accurately simulate the reaction process, smaller time steps may be required than that required to accurately simulate advection or the diffusion processes. This is particularly true for groundwater problems.

Operator splitting permits the solution of the nonlinear ordinary differential equations using time steps much smaller than that required to accurately simulate advection or diffusion. This results in considerable savings in computer resources by avoiding the unnecessary solution of advection and diffusion for many intermediate time steps that would occur in single step schemes. In addition, optimum solution techniques can be employed for each process.

Time-splitting algorithms are also more suited for implementation on parallel computers, where advection and diffusion can be treated in parallel. The reaction source-sink ordinary differential

Numerical Methods fo r Solving the One-Dimensional Advective-Diffusion Equation 2.12

equation can also be solved in parallel on a number of processors. The implementation of time­ splitting using parallel processors has been discussed by Wheeler and Dawson (1987).

Other situations where the use of time-splitting may lead to a more efficient solution than single time stepping have been discussed by LeVeque and Öliger (1983).

The time-splitting procedure will be described for the solution of the linear advective-diffusion equation without a source-sink and reaction term

dc

T t (2.13)

There are a number of approaches that can be used to construct time-splitting schemes with a desired accuracy in time. Amongst these are the method proposed by D’Yakonov (1963) and that due to Strang (1968). Only the Strang splitting approach is used here.

Equation (2.13) can be written in the following form

— = C (x,t,d/dx,d2/dx2)c (2.14)

dt

where C is a linear operator independent of the temporal derivative.

For equation (2.13) C involves two operators, C — LA LD, where LA = -ud/dx, is the advection operator and LD = Dd2!d^, is the diffusion operator.

The exact solution to equation (2.14) for any interior computational node at time t + At, given the solution at time t can be expressed by the following Taylor series

71 + 1

<7 A

2 dx2

n

Cj.

Substituting the differential equation, equation (2.14), then

< f ‘ = c," + At(L, + Ld)c" + ^ - ( L l + ( L / 0 + V J + L l)C; + • • •

= exp(At(LA + Ld))c- = exp(AtC)Cj.

(2.15)

Equation (2.15) represents a strategy for solving equation (2.13).

The various time-splitting schemes approximate exp(At(LA + LD)) to the desired accuracy. For example, equation (2.15) can be approximated by

c;+1 « exp{AtLA)exp{AtLD)c*. (2.16)

The time-splitting strategy for approximating c;n+1 is to solve the diffusion operator

conditions c / for the solution of the advection operator exp(AtLA)Cjn. Although the order of applying the two processes is unimportant, higher accuracy is achievable by alternating the process, that is the order of splitting is reversed at each time step (see, for example Valocchi and Malmstead [1992]).

The order of accuracy of equation (2.16) can be established using Taylor’s series, where c f 1 - c/ + At(LÄ + Ld) + + 2 L /d + * 0 (A t3).

Comparing this equation with equation (2.15) reveals that the approximation is only first-order accurate if LA and LD do not commute, LALD ^ L ^ , otherwise it is exact.

Alternative higher-order time-stepping approximations have been proposed. In many of these schemes the sequence of operations is symmetric over a single time step. Consider the following approximation

Cj+l * ^-(exp(AtLA)exp(AtLD) + expiAtL^expiAtLjjc". (2.17)

The second-order accuracy of this scheme is obtained from the symmetry of the operations. Unfortunately this scheme requires twice the computational effort per time step than the first- order scheme, equation (2.16).

Strang (1968) described a second-order scheme based on the following splitting operator

n + i

Cj ~ exp exp(AtLA)exp (2.18)

which is more efficient that the solution of equation (2.17). Again the order of the operations in not important.

This scheme is almost an alternating solution of exp(AtLA) and exp(AtLD) over At. The combination of exp(Atl2L]D) at the end of one time step and again at the beginning of the next can be replaced by the single operator exp{AtLD) without affecting the accuracy of the scheme. Only at the first step and when the solution is required do the half time steps enter. Therefore equation (2.18) can be replaced by

n + 1

Cj ~ exp

2 Ld exp(AtLA)exp(AtLD) • • • exp(AtLD)exp(AtLA)exp

*L

(2.19)

which may simplify the implementation of the procedure and in certain circumstances may lead to a more efficient solution of the original problem (see, for example LeVeque and Öliger [1983]).

Stability analysis for time-splitting schemes does not necessarily follow from the stability of each of the steps unless the amplification factors commute. For certain problems, LeVeque and Öliger (1983) and Sheng (1989) have illustrated that although each operation may be stable, the product

Numerical Methods fo r Solving the One-Dimensional Advective-Diffusion Equation 2.14

operator may not.

The operator splitting scheme is stable provided that each independent operator is stable (see, for example LeVeque and Öliger [1983] and Noye [1987]). Noye (1987) showed that in general the two-time step splitting methods for solving the one-dimensional advective-diffusion equation have a larger stability region than the equivalent one-time step direct scheme. However, they may require twice the computational effort compared to the one-time step methods. In addition, whenever the component finite difference scheme used in the first time step is explicit and that of the second is either explicit or implicit, the equivalent one-time step finite difference scheme has the same stability range as the time-split procedures involving the two component finite difference schemes applied over half steps (Noye [1987]).

Noye (1987) also showed that the local truncation error of a time-splitting scheme is equal to the sum of those of the individual components.

Time-splitting can also be used for simulating discontinuous solutions of scalar conservative laws. Many of the desirable features of conservative finite difference schemes are preserved in the splitting process. Crandall and Majda (1980) discuss in detail the use of time-splitting for approximating discontinuous solutions. Beale and Majda (1981) established the error estimates of splitting algorithms for the solution of nonlinear equations.

A significant difficulty associated with time-splitting schemes is in determining the appropriate boundary conditions for each of the steps. Improper boundary conditions can seriously degrade the accuracy of the solution. A method for deriving boundary conditions for time-splitting schemes has been given by LeVeque and Öliger (1983) and Noye (1987).

There is no restriction on the numerical scheme that can be used to solve the advection or diffusion operators. Any combination of stable methods can be used for the separate steps. A number of suitable numerical schemes that could be used in a time-splitting scheme for modelling advection and diffusion of a smooth profile and a step function will be discussed below.