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Why do we need Numerical Methods?

Why do we need Numerical Methods?. Conceptual model. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication. Conceptual model. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication. Conceptual model. Equilibrium Equations

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Why do we need Numerical Methods?

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  1. Why do we need Numerical Methods?

  2. Conceptual model

  3. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model

  4. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Equilibrium Equations Heuristics Simplifications Continuous Discrete Experiments, Statistics, Prediction Mathematical Model

  5. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Equilibrium Equations Heuristics Simplifications Continuous Discrete Experiments, Statistics, Predictions Mathematical Model other nonlinear ODE linear

  6. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Mathematical Model other nonlinear ODE linear Ax=b Find A-1 x=A-1b ?????

  7. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Ax=b Find A-1 x=A-1b ?????

  8. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ?????

  9. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Optimal control, PDEs, DAEs,… Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ?????

  10. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Optimal control PDEs, DAEs,… Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ????? Numerical Methods

  11. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Optimal control, PDEs, DAEs,… Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ????? Numerical Methods Computed Solution

  12. Numerical Methods Computed Solution • Algorithms • Programming • Convergence • Error • Analysis • Stability • What can go wrong? • Which algorithm works best • on which type of problem?

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