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Time-stepping methods for the wave eq. (Supp) (Setup (Ch0) (2nd order in…
Time-stepping methods for the wave eq. (Supp)
Time stepping methods (Ch2)
Methods for 1st order system (Ch2.2)
conservation of energy
wave eq.
mixed var. form.
reduce to 1st order system of PDEs
space discretization -> ODE system
skew symmetric matrix
Hamiltonian structure
\( \Rightarrow \) EVa imaginary
symplectic methods
symplectic Euler method
Newmark for 2nd order (Ch2.1)
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
Setup (Ch0)
2nd order in time \( \ddot{u}+Au=f \)
ICs for \( u(0), \dot{u}(0) \)
\( A \) is symm, elliptic operator
Examples (Ch1)
scalar wave eq. (acoustic)
electromagnetic wave eq.
elastic wave eq.