Modeling of geochemical processes
                                   Numeric Mathematics Refreshment

                                             J. Faimon

                                 Modeling of geochemical processes
                                        Numeric mathematics

Numeric mathematics

Search of solution by a substitution of exact numerals for variables.

Large number of repeated numeric operation  

    Personal computers 



Iterations and algorithms 

*      algorithm -- an instruction how to solve the given task

*      iteration algorithm -- technique of calculation in the repetitious steps, at which the result
from the previous step is input into next step[1].

*      iteration -- the step in the iteration algorithm

      [1] Algorithm should "converge" (it should approach solution).

                                 Modeling of geochemical processes
                                        Numeric mathematics

Example: Calculation of square root               



It is a solution of the equation                ,  for x

We have to find, e.g., square root of 16, N = 16.



The arrangement of the definitional equation gives                 .   

 Let us calculate x.



At first (zero step), we substitute for x on the right hand of the equation its estimation
(arbitrary number), e.g., x[0] = 2, and calculate new value of x, i.e., x[1].  We substitute the
found solution for x and repeat the calculation till the solution is not constant. 



The solution procedure (algorithm) can be written as

                                                                                                                  

               where k is step number; iteration number (we start by zero step)

                                 Modeling of geochemical processes
                                        Numeric mathematics

                                                                                        Calculation:



 As can be seen, the algorithm does not converge. It must be revised!

                                 Modeling of geochemical processes
                                        Numeric mathematics

Algorithm formulation

   Solution of non-linear

   equations: Newton method

                                 Modeling of geochemical processes
                                        Numeric mathematics

Example:



Find the roots of the equation

        x^3 -- 4x^2 + 5x -- 2 =  0.

 The derivation of f(x) for x is  

           3x^2 - 8x + 5









   The roots are 

          x[1] = 2,  x[2,3] = 1

                                 Modeling of geochemical processes
                                        Numeric mathematics

Example: Equilibrium in the open system  calcite-H[2]O-CO[2

]system: CaCO[3(s)], CO[2], H[2]O, Ca^2+, HCO[3]^-, CO[3]^2-, OH^-, H^+, H[2]CO[3]  (9 components)

activity coefficient ~ 1,   a ~ mol/l,           

4 components are given: activity of CaCO[3(s)] and H[2]O, p[CO2] =  const = 3.10^-4 atm    and
[H[2]CO[3]] = const = K[H] p[CO2

]5 variables:  x[1] = [H^+],  x[2] = [OH^-],  x[3] = [Ca^2+],  x[4] = [HCO[3]^-],  x[5] =
[CO[3]^2-]  

5 equations: (1) electro-neutrality

and equilibrium equations for:

(2) H[2]O, (3) calcite-H[2]O system, carbonate dissociation into (4) first and (5) second stage!

5 variables in 5 functions:

f[1](x) = x[1] - x[2] + 2 x[3] - x[4] -- 2 x[5] = 0         (electro-neutrality)

f[2](x) = x[1] x[2] -- K[w] = 0                              (water ion product)

f[3](x) = x[1] x[5]/x[4] -- K[2] = 0                          (dissociation constant K[2])

f[4](x) = x[1] x[4] -- K[1] K[H]  p[CO2] = 0                (dissociation constant K[1])

f[5](x) = x[3] x[5] -- K[s] = 0                               (calcite dissolution product)