1: Chapters
- Page ID
- 108080
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Introduction
- This page covers solving first-order ordinary differential equations (ODEs), focusing on initial value problems (IVP) where unique solutions depend on initial conditions. It explains sampled functions for representing continuous data as discrete points, which is essential for numerical methods.
- 1.2: Forward Euler method
- This page covers the Forward Euler method for solving first-order ordinary differential equations (ODEs), focusing on its implementation, error estimation (local truncation and global error), and limitations in accuracy and stability, particularly for exponential growth and oscillatory problems like the simple harmonic oscillator. The stability conditions are discussed, emphasizing the influence of step size on numerical solutions.
- 1.3: Backward Euler method
- This page covers the backward Euler method as an ODE solver, emphasizing its implicit nature and reliance on root-finding algorithms for future value computation. It explores applications to the logistic equation and simple harmonic oscillator, noting potential oscillatory decay and stability issues, particularly within exponential contexts. The method's stability is heavily influenced by parameter values and may require smaller step sizes than forward Euler to prevent instability.
- 1.4: Predictor-corrector methods and Runge-Kutta
- This page covers numerical methods for solving ordinary differential equations (ODEs), highlighting Heun's method, the midpoint method, and the fourth-order Runge-Kutta method (RK4). Heun's method improves accuracy with a predictor-corrector approach, achieving \(O(h^2)\) error. The midpoint method also offers enhanced accuracy, while RK4 provides greater precision with \(O(h^4)\) error.
- 1.5: Adaptive stepping
- This page covers the significance of choosing an appropriate stepsize \(h\) for solving ordinary differential equations (ODEs) to balance accuracy and runtime. It emphasizes adaptive stepping methods that adjust \(h\) based on error estimates, illustrated by examples like the logistic and Van der Pol equations. The approach enhances computational efficiency, maintaining accuracy in sensitive regions while speeding up calculations in less sensitive areas.
- 1.6: Linear multistep methods
- This page covers alternative methods for solving first-order ordinary differential equations (ODEs), including the trapezoidal and Adams-Bashforth methods, both achieving \(O(h^2)\) accuracy. It further explores the third-order Adams-Bashforth, using quadratic interpolation for better accuracy (\(O(h^3)\), and contrasts it with the implicit Adams-Moulton method that offers stability with larger step sizes.
- 1.7: Symplectic integrators
- This page covers symplectic integrators, particularly the symplectic Euler method, which is vital for accurately simulating mechanical systems like harmonic oscillators and celestial bodies by conserving energy. It contrasts symplectic methods with general ODE solvers, explaining energy preservation through Hamiltonians and the importance of meeting specific conditions.
- 1.8: Review and summary
- This page outlines ordinary differential equation (ODE) solvers, distinguishing between explicit and implicit methods, with a focus on their computation techniques. It highlights the importance of stepsize for accuracy and differentiates between one-step and multistep methods, including adaptive step size strategies. Additionally, it introduces symplectic integrators for energy conservation and offers Matlab programs for practical application.