Completed notes of the course Numerical Methods

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Numerical Methods Course held by Prof. Christian Vergara Year 2020-2021 Notes by Alessandro M. Ippoliti Leonhard Euler 1707-1783. 2 Contents I Finite Differences 5 1 General Overview 7 2 ODE Approximations 9 2.1 Approximations of the 1 st Derivative . . . . . . . . . . . . . . 9 2.1.1 Forward Euler Method . . . . . . . . . . . . . . . . . . 9 2.1.2 Backward Euler Method . . . . . . . . . . . . . . . . . 10 2.1.3 Central Approximation Formula . . . . . . . . . . . . . 10 2.1.4 Forward and Backward 2 nd Order Formulas . . . . . . 12 2.1.5 Approximations of the 2 nd Derivative . . . . . . . . . . 12 2.2 Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . 12 2.3 Model Evaluation Parameters . . . . . . . . . . . . . . . . . . 14 2.3.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.2 Zero Stability . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.4 Absolute Stability . . . . . . . . . . . . . . . . . . . . . 17 2.4 The Theta Method . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 The Advection Problem . . . . . . . . . . . . . . . . . . . . . 19 2.6 Solution through the Forward Euler Centered Method . . . . . 20 2.6.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6.2 Zero Stability . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.4 Absolute Stability . . . . . . . . . . . . . . . . . . . . . 23 2.7 The Upwind Method . . . . . . . . . . . . . . . . . . . . . . . 24 2.7.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7.2 Absolute Stability . . . . . . . . . . . . . . . . . . . . . 25 2.8 Numerical Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 26…