_ _ ] _ _ _ _ _ [ ˆ„(x, t) ˆt + Èv(x, t), Ò x „(x, t)Í ≠ H ú(x, t, v(x, t)) = ‘ x „(x, t), x œ , t > 0, ˆfl(x, t) ˆt + Ò x · (fl(x, t)v(x, t)) + ‘ x fl(x, t) = 0, x œ , t > 0, v(x, t) = Ò pH(x, t, Ò x „(x, t)), x œ , t > 0, „(x, 0) = g(x), fl(x, T) = c, x œ , (5) In PDHG method, the updates for „ and fl are related to the first (linear wrt „) and second line (linear wrt fl), respectively. We use implicit time discretization We can have a larger CFL number ( t x max p ÎÒ pH(x, t, p)Î), e.g., in example 2, CFL number is 0.036 for explicit scheme, but it can be about 10.8 in our method. Meng, Hao, Liu, Yang, Li, Osher PDHG HJ Aug 10, 2023 4 / 8 36