LectureNotesOutline (6).pdf - Outline Notes AMME2000 BMET2960 B Thornber April 9 2020 Contents 1 Introduction to Numerical Methods 3 1.1 Why do we need

LectureNotesOutline (6).pdf - Outline Notes AMME2000...

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Outline Notes AMME2000 & BMET2960 B. Thornber April 9, 2020 Contents 1 Introduction to Numerical Methods 3 1.1 Why do we need numerical methods? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Interpolation, Numerical Integration and Differentiation . . . . . . . . . . . . . . . . . . . 4 1.2.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Taylor Series Derivation of the Central Difference Scheme . . . . . . . . . . . . . . . . . 6 2 Partial Differential Equations 8 3 Heat Equation 9 3.1 Introduction and Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Expected Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Use of the Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Simple Inhomogeneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 More Complex Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5.1 Step 1: Steady State solution T ss ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5.2 Step 2: Formulation and Solution of the Homogeneous Problem . . . . . . . . . . 13 3.5.3 Step 3: Fourier Series for non-integer modes . . . . . . . . . . . . . . . . . . . . 14 3.5.4 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Key Concepts in Numerical Methods 16 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Order Of Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1
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4.5 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.6 Quantifying the Errors with Error Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.7 Computing the Observed Order of Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 Numerical Solution of the Heat Equation 19 5.1 Forward in Time, Central in Space (FTCS) . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.1.1 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.1.2 von Neumann Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Backward in Time, Central in Space Scheme (BTCS) . . . . . . . . . . . . . . . . . . . . 22 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6 The Wave Equation 24 6.1 Derivation of the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.3 Physical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7 Numerical Solution of the Wave Equation 27 7.1 Second Order Accurate Central Difference in Time and Space . . . . . . . . . . . . . . . 27 7.2 von Neumann Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.3 Advanced - Special Case of a Uniform Grid . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 8 Laplace and Poisson Equations 31 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.2 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9 Numerical Solution of the Laplace and Poisson Equation 35 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.2 Second Order Central Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A Derivation of the Heat Equation 38 B Derivation of the Fourier Sine Series Coefficients 39 2
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Introduction These outlines should be read in conjunction with the lecture slides and online videos for AMME2000 & BMET2960. 1 Introduction to Numerical Methods 1.1Why do we need numerical methods?
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