Time stepping methods (Ch2)
Setup (Ch0)
Examples (Ch1)
2nd order in time ¨u+Au=f
scalar wave eq. (acoustic)
Methods for 1st order system (Ch2.2)
Newmark for 2nd order (Ch2.1)
ICs for \( u(0), \dot{u}(0) \)
\( A \) is symm, elliptic operator
electromagnetic wave eq.
elastic wave eq.
conservation of energy
wave eq.
skew symmetric matrix
symplectic Euler method
ODE \( M\ddot{u}+Ku=f \)
based on taylor expansion for \( u, \ddot{u}\)
satsifies discrete energy conservation
equivalent stiffness matrix \( K_{eq} \)
pos. def. \( \Rightarrow \) conservation proves stability
single step methods
solve lin. system w/ spd matrix
conditional (=max. time-step)/unconditional stability
mixed var. form.
reduce to 1st order system of PDEs
Hamiltonian structure
\( \Rightarrow \) EVa imaginary
space discretization -> ODE system
symplectic methods