DIGITAL SIMULATION IN ELECTROCHEMISTRY
CONTENTS INTRODUCTION BASIC EQUATIONS GENERAL SOME MATHEMATICS: TRANSPORT EQUATIONS DI?USION DI?USION CURRENT CONVECTION MIGRATION TOTAL TRANSPORT EQUATION HOMOGENEOUS KINETICS HETEROGENEOUS KINETICS NORMALISATION – MAKING THE VARIABLES DIMENSIONLESS SOME MODEL SYSTEMS AND THEIR NORMALISATIONS POTENTIAL STEPS CONSTANT CURRENT LINEAR SWEEP VOLTAMMETRY (LSV) ADSORPTION KINETICS APPROXIMATIONS TO DERIVATIVES APPROXIMATION ORDER TWO-POINT FIRST DERIVATIVE APPROXIMATIONS MULTI-POINT FIRST DERIVATIVE APPROXIMATIONS THE CURRENT APPROXIMATION THE CURRENT APPROXIMATION FUNCTION G HIGH-ORDER COMPACT (HERMITIAN) CURRENT APPROXIMATION SECOND DERIVATIVE APPROXIMATIONS DERIVATIVES ON UNEVENLY SPACED POINTS ERROR ORDERS A SPECIAL CASE CURRENT APPROXIMATION A SPECI?C APPROXIMATION ORDINARY DI?ERENTIAL EQUATIONS AN EXAMPLE ODE LOCAL AND GLOBAL ERRORS WHAT DISTINGUISHES THE METHODS EULER METHOD RUNGE-KUTTA, RK BACKWARDS IMPLICIT, BI TRAPEZIUM OR MIDPOINT METHOD BACKWARD DI?ERENTIATION FORMULA, BDF STARTING BDF EXTRAPOLATION 4 10 KIMBLE & WHITE, KW 4 10 1 USING KW AS A START FOR BDF 4 11 SYSTEMS OF ODES 4 12 ROSENBROCK METHODS 4 12 1 APPLICATION TO A SIMPLE EXAMPLE ODE 4 12 2 ERROR ESTIMATES THE EXPLICIT METHOD THE DISCRETISATION PRACTICALITIES CHRONOAMPEROMETRY AND -POTENTIOMETRY HOMOGENEOUS CHEMICAL REACTIONS (HCR) THE REACTION LAYER LINEAR SWEEP VOLTAMMETRY BOUNDARY CONDITION HANDLING BOUNDARY CONDITIONS CLASSI?CATION OF BOUNDARY CONDITIONS SINGLE SPECIES: THE U-V DEVICE DIRICHLET CONDITION DERIVATIVE BOUNDARY CONDITIONS TWO SPECIES TWO-POINT DERIVATIVE CASES TWO SPECIES WITH COUPLED REACTIONS U-V BRUTE FORCE A GENERAL FORMALISM UNEQUAL INTERVALS TRANSFORMATION DISCRETISING THE TRANSFORMED EQUATION THE CHOICE OF PARAMETERS DIRECT APPLICATION OF AN ARBITRARY GRID CHOICE OF PARAMETERS CONCLUDING REMARKS ON UNEQUAL SPATIAL INTERVALS UNEQUAL TIME INTERVALS IMPLEMENTATION OF EXPONENTIALLY INCREASING TIME INTERVALS ADAPTIVE INTERVAL CHANGES SPATIAL INTERVAL ADAPTATION TIME INTERVAL ADAPTATION THE COMMONLY USED IMPLICIT METHODS THE LAASONEN METHOD OR BI THE CRANK-NICOLSON METHOD, CN SOLVING THE IMPLICIT SYSTEM USING FOUR-POINT SPATIAL SECOND DERIVATIVES IMPROVEMENTS ON CN AND LAASONEN DAMPING THE CN OSCILLATIONS MAKING LAASONEN MORE ACCURATE HOMOGENEOUS CHEMICAL REACTIONS NONLINEAR EQUATIONS COUPLED EQUATIONS OTHER METHODS THE BOX METHOD IMPROVEMENTS ON STANDARD METHODS THE KIMBLE AND WHITE METHOD MULTI-POINT SECOND SPATIAL DERIVATIVES DUFORT-FRANKEL SAUL’YEV HOPSCOTCH RUNGE-KUTTA HERMITIAN METHODS METHOD OF LINES (MOL) AND DI?ERENTIAL ALGEBRAIC EQUATIONS (DAE) THE ROSENBROCK METHOD AN EXAMPLE, THE BIRK-PERONE SYSTEM FEM, BEM AND FAM (BRIE?Y) ORTHOGONAL COLLOCATION, OC CURRENT CALCULATION WITH OC A NUMERICAL EXAMPLE EIGENVALUE-EIGENVECTOR METHOD INTEGRAL EQUATION METHOD THE NETWORK METHOD 9 10 TREANOR METHOD 9 11 MONTE CARLO METHOD ADSORPTION 10 1 TRANSPORT AND ISOTHERM LIMITED ADSORPTION 10 2 ADSORPTION RATE LIMITED ADSORPTION E?ECTS DUE TO UNCOMPENSATED RESISTANCE AND CAPACITANCE 11 1 BOUNDARY CONDITIONS 11 1 1 AN EXAMPLE TWO-DIMENSIONAL SYSTEMS 12 1 THEORIES 12 1 1 THE ULTRAMICRODISK ELECTRODE, UMDE 12 1 2 OTHER MICROELECTRODES 12 1 3 SOME RELATIONS 12 2 SIMULATIONS 12 3 SIMULATING THE UMDE 12 3 1 DIRECT DISCRETISATION 12 3 2 DISCRETISATION IN THE MAPPED SPACE 12 3 3 A REMARK ON THE BOUNDARY CONDITIONS 13 1 SOME FLUID DYNAMICS 13 1 1 LAYER RELATIONS 13 2 ELECTRODES IN FLOW SYSTEMS 13 3 SIMULATIONS 13 4 A SIMPLE EXAMPLE: THE BAND ELECTRODE IN A CHANNEL FLOW 13 5 NORMALISATIONS PERFORMANCE 14 1 CONVERGENCE 14 2 CONSISTENCY 14 3 STABILITY 14 3 1 HEURISTIC METHOD 14 3 2 VON NEUMANN STABILITY ANALYSIS 14 3 3 MATRIX STABILITY ANALYSIS 14 3 4 SOME SPECIAL CASES 14 4 THE STABILITY FUNCTION 14 5 ACCURACY ORDER 14 5 1 ORDER DETERMINATION 14 6 ACCURACY, E?CIENCY AND CHOICE 14 7 SUMMARY OF METHODS PROGRAMMING 15 1 LANGUAGE AND STYLE 15 2 DEBUGGING 15 3 LIBRARIES SIMULATION PACKAGES