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NumMeth_Handout_5 - Numerical Methods in Chemical...

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Numerical Methods in Chemical Engineering 5 Solving systems of non-linear equations 5 Solving systems of non-linear equations ....................................................... 1 5.1 Overview .................................................................................................. 2 5.2 Passing functions ...................................................................................... 3 5.3 1D Newtons Method (something you did at school) ............................... 4 5.4 Newton's method in more than one dimension ........................................ 6 5.4.1 Solving the system of two coupled equations ................................... 9 5.4.2 Solution trajectories ........................................................................ 11 5.4.3 The reduced Newton step method ................................................... 13 5.4.4 Problems with Newton's Method .................................................... 14 5.5 A chemical engineering example - equilibrium .................................... 16 5.5.1 The element balance equations ....................................................... 17 5.5.2 The equilibrium equations ............................................................... 17 5.5.3 Thermodynamic data ....................................................................... 18 5.5.4 Solving the equations ..................................................................... 19 5.5.5 Full solution to the equilibrium problem in Matlab ........................ 21 5.6 Convergence of the Newton-method ..................................................... 23 5.7 Calculating the Jacobian Matrix ............................................................ 24 5.7.1 Finite differences ............................................................................. 24 5.7.2 Broydens Method ............................................................................ 25 5.7.3 Taking advantage of sparsity .......................................................... 26 5.8 Trust region method (Used by fsolve) ................................................... 27 5.9 Summary ................................................................................................ 28 5.10 Amendments ....................................................................................... 29 5.10.1 Updated version of Newton1D .................................................... 29 5.10.2 Updated version of Newton ......................................................... 30 - 1 -
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