*J. G. Hayes*

- Published in print:
- 2005
- Published Online:
- January 2008
- ISBN:
- 9780198565932
- eISBN:
- 9780191714016
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198565932.003.0011
- Subject:
- Mathematics, History of Mathematics

This chapter presents an overview of the programming process for the Pilot ACE. This programme was for the back-substitution phase of solving a set of linear algebraic equations, with multiple ...
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This chapter presents an overview of the programming process for the Pilot ACE. This programme was for the back-substitution phase of solving a set of linear algebraic equations, with multiple right-hand sides, up to order 32. Topics covered include storage, two stages of programming, programming back substitution, steps of the programme, and multiple right-hand sides.Less

This chapter presents an overview of the programming process for the Pilot ACE. This programme was for the back-substitution phase of solving a set of linear algebraic equations, with multiple right-hand sides, up to order 32. Topics covered include storage, two stages of programming, programming back substitution, steps of the programme, and multiple right-hand sides.

*James C. G. Walker*

- Published in print:
- 1991
- Published Online:
- November 2020
- ISBN:
- 9780195045208
- eISBN:
- 9780197560020
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195045208.003.0005
- Subject:
- Earth Sciences and Geography, Geochemistry

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in ...
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The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. Most simulations of environmental change involve several interacting reservoirs. In this chapter I shall explain how to apply the numerical scheme described in the previous chapter to a system of coupled equations. Figure 3-1, adapted from Broecker and Peng (1982, p. 382), is an example of a coupled system. The figure presents a simple description of the general circulation of the ocean, showing the exchange of water in Sverdrups (1 Sverdrup = 106 m3/sec) among five oceanic reservoirs and also the addition of river water to the surface reservoirs and the removal of an equal volume of water by evaporation. The problem is to calculate the steady-state concentration of dissolved phosphate in the five oceanic reservoirs, assuming that 95 percent of all the phosphate carried into each surface reservoir is consumed by plankton and carried downward in particulate form into the underlying deep reservoir. The remaining 5 percent of the incoming phosphate is carried out of the surface reservoir still in solution.
Less

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. Most simulations of environmental change involve several interacting reservoirs. In this chapter I shall explain how to apply the numerical scheme described in the previous chapter to a system of coupled equations. Figure 3-1, adapted from Broecker and Peng (1982, p. 382), is an example of a coupled system. The figure presents a simple description of the general circulation of the ocean, showing the exchange of water in Sverdrups (1 Sverdrup = 106 m3/sec) among five oceanic reservoirs and also the addition of river water to the surface reservoirs and the removal of an equal volume of water by evaporation. The problem is to calculate the steady-state concentration of dissolved phosphate in the five oceanic reservoirs, assuming that 95 percent of all the phosphate carried into each surface reservoir is consumed by plankton and carried downward in particulate form into the underlying deep reservoir. The remaining 5 percent of the incoming phosphate is carried out of the surface reservoir still in solution.