WARNING: Preliminary version (expect typos!)
Overview of truncation error analysis
Abstract problem setting
Error measures
Truncation errors in finite difference formulas
Example: The backward difference for \( u'(t) \)
Example: The forward difference for \( u'(t) \)
Example: The central difference for \( u'(t) \)
Overview of leading-order error terms in finite difference formulas
Software for computing truncation errors
Truncation errors in exponential decay ODE
Truncation error of the Forward Euler scheme
Truncation error of the Crank-Nicolson scheme
Truncation error of the \( \theta \)-rule
Using symbolic software
Empirical verification of the truncation error
Increasing the accuracy by adding correction terms
Extension to variable coefficients
Exact solutions of the finite difference equations
Computing truncation errors in nonlinear problems
Truncation errors in vibration ODEs
Linear model without damping
The truncation error of a centered finite difference scheme
The truncation error of approximating \( u'(0) \)
Truncation error of the equation for the first step
Computing correction terms
Model with damping and nonlinearity
Extension to quadratic damping
The general model formulated as first-order ODEs
The forward-backward scheme
A centered scheme on a staggered mesh
Truncation errors in wave equations
Linear wave equation in 1D
Finding correction terms
Extension to variable coefficients
1D wave equation on a staggered mesh
Linear wave equation in 2D/3D
Truncation errors in diffusion equations
Linear diffusion equation in 1D
The Forward Euler scheme in time
The Crank-Nicolson scheme in time
Linear diffusion equation in 2D/3D
A nonlinear diffusion equation in 2D
Exercises
Exercise 1: Truncation error of a weighted mean
Exercise 2: Simulate the error of a weighted mean
Exercise 3: Verify a truncation error formula
Exercise 4: Truncation error of the Backward Euler scheme
Exercise 5: Empirical estimation of truncation errors
Exercise 6: Correction term for a Backward Euler scheme
Exercise 7: Verify the effect of correction terms
Exercise 8: Truncation error of the Crank-Nicolson scheme
Exercise 9: Truncation error of \( u'=f(u,t) \)
Exercise 10: Truncation error of \( [D_t D_tu]^n \)
Exercise 11: Investigate the impact of approximating \( u'(0) \)
Exercise 12: Investigate the accuracy of a simplified scheme