$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\uexd}[1]{{u_{\small\mbox{e}, #1}}} \newcommand{\vex}{{v_{\small\mbox{e}}}} \newcommand{\vexd}[1]{{v_{\small\mbox{e}, #1}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\tp}{\thinspace .} \newcommand{\Ddt}[1]{\frac{D #1}{dt}} \newcommand{\E}[1]{\hbox{E}\lbrack #1 \rbrack} \newcommand{\Var}[1]{\hbox{Var}\lbrack #1 \rbrack} \newcommand{\Std}[1]{\hbox{Std}\lbrack #1 \rbrack} \newcommand{\xpoint}{\boldsymbol{x}} \newcommand{\normalvec}{\boldsymbol{n}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\x}{\boldsymbol{x}} \newcommand{\X}{\boldsymbol{X}} \renewcommand{\u}{\boldsymbol{u}} \renewcommand{\v}{\boldsymbol{v}} \newcommand{\w}{\boldsymbol{w}} \newcommand{\V}{\boldsymbol{V}} \newcommand{\e}{\boldsymbol{e}} \newcommand{\f}{\boldsymbol{f}} \newcommand{\F}{\boldsymbol{F}} \newcommand{\stress}{\boldsymbol{\sigma}} \newcommand{\strain}{\boldsymbol{\varepsilon}} \newcommand{\stressc}{{\sigma}} \newcommand{\strainc}{{\varepsilon}} \newcommand{\I}{\boldsymbol{I}} \newcommand{\T}{\boldsymbol{T}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\ii}{\boldsymbol{i}} \newcommand{\jj}{\boldsymbol{j}} \newcommand{\kk}{\boldsymbol{k}} \newcommand{\ir}{\boldsymbol{i}_r} \newcommand{\ith}{\boldsymbol{i}_{\theta}} \newcommand{\iz}{\boldsymbol{i}_z} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\Iz}{\mathcal{I}_z} \newcommand{\It}{\mathcal{I}_t} \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifd}{{I_d}} % for FEM \newcommand{\Ifb}{{I_b}} % for FEM \newcommand{\setb}[1]{#1^0} % set begin \newcommand{\sete}[1]{#1^{-1}} % set end \newcommand{\setl}[1]{#1^-} \newcommand{\setr}[1]{#1^+} \newcommand{\seti}[1]{#1^i} \newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}} \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\refphi}{\tilde\basphi} \newcommand{\psib}{\boldsymbol{\psi}} \newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)} \newcommand{\xno}[1]{x_{#1}} \newcommand{\Xno}[1]{X_{(#1)}} \newcommand{\yno}[1]{y_{#1}} \newcommand{\Yno}[1]{Y_{(#1)}} \newcommand{\xdno}[1]{\boldsymbol{x}_{#1}} \newcommand{\dX}{\, \mathrm{d}X} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} \newcommand{\Real}{\mathbb{R}} \newcommand{\Integerp}{\mathbb{N}} \newcommand{\Integer}{\mathbb{Z}} $$

Overview of what truncation errors are
Abstract problem setting
Various error measures
Truncation errors in finite difference formulas
Example: The backward difference for $ u'(t) $
Taylor series
Taylor series inserted in the backward difference approximation
The forward difference for $ u'(t) $
The central difference for $ u'(t) $ (1)
The central difference for $ u'(t) $ (1)
Leading-order error terms in finite differences (1)
Leading-order error terms in finite differences (2)
Leading-order error terms in mean values (1)
Leading-order error terms in mean values (2)
Software for computing truncation errors
Symbolic computing with difference operators
Truncation errors in exponential decay ODE
Truncation error of the Forward Euler scheme
Truncation error of the Crank-Nicolson scheme
Test the understanding!
Truncation error of the $ \theta $-rule
Using symbolic software
Empirical verification of the truncation error (1)
Empirical verification of the truncation error (2)
Empirical verification of the truncation error in the Forward Euler scheme
Empirical verification of the truncation error in the Forward Euler scheme
Increasing the accuracy by adding correction terms
Lowering the order of the derivative in the correction term
With a correction term Forward Euler becomes Crank-Nicolson
Correction terms in the Crank-Nicolson scheme (1)
Correction terms in the Crank-Nicolson scheme (2)
Extension to variable coefficients
Exact solutions of the finite difference equations
Computing truncation errors in nonlinear problems (1)
Computing truncation errors in nonlinear problems (2)
Truncation errors in vibration ODEs
Linear model without damping
Truncation errors in the initial condition
Computing correction terms
Model with damping and nonlinearity
Carrying out the truncation error analysis
Extension to quadratic damping
The truncation error for quadratic damping (1)
The truncation error for quadratic damping (2)
The general model formulated as first-order ODEs
The forward-backward scheme
Truncation error analysis
A centered scheme on a staggered mesh
Truncation error analysis (1)
Truncation error analysis (2)

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