Numerical methods have been developed in order to give solutions to problems where there is no analyticalsolution. An analytical solution is the answer to a given problem which when evaluated is...

Numerical methods have been developed in order to give solutions to problems where there is no analytical solution. An analytical solution is the answer to a given problem which when evaluated is exact. An analytical solution may be very complex and require a computer to solve, but is exact up to the precision of the method used to evaluate it - on a computer this is typically 8, 16 or 32 significant figures (single, double or quadruple precision). An example of an analytical solution could be the derivative of f(x) = sin(x), which is:


derivations.dvi 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 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 . . . . . . . . . . . . . . . . . . . . .