Numerical Methods For Conservation Laws From Analysis To Algorithms __full__ Official

where ( F^low ) is a first-order monotone flux (e.g., Godunov), ( F^high ) is a high-order linear flux (e.g., Lax-Wendroff), and ( \phi(r) ) is a limiter function depending on the ratio of consecutive gradients ( r = (U_i - U_i-1)/(U_i+1 - U_i) ).

For problems with fine-scale structures (turbulence, acoustics), second-order accuracy is insufficient. Enter and Weighted ENO (WENO) schemes (Harten et al., 1987; Liu, Osher, Chan, 1994; Jiang & Shu, 1996). where ( F^low ) is a first-order monotone flux (e

While high-order schemes (like Simpson’s rule equivalents) are fast and sharp, they tend to create artificial "wiggles" or oscillations near shocks (Godunov’s Theorem). Jiang & Shu

[ F_i+1/2 = F_i+1/2^low + \phi(r)(F_i+1/2^high - F_i+1/2^low) ] especially in seismology (SPECFEM) and aeroacoustics.

DG has revolutionized computational physics, especially in seismology (SPECFEM) and aeroacoustics.