- #1
mertcan
- 345
- 6
Hi PF, initially I would like you to focus on that link https://books.google.com.tr/books?id=Dkp6CwAAQBAJ&pg=PA389&lpg=PA389&dq=runge+kutta+method++is+tvd+proof&source=bl&ots=47ULQDVwcC&sig=e2zjdnXENJ7WxBbrf6hXkSouvLI&hl=tr&sa=X&ved=0ahUKEwjU5Z2XsbXZAhUMCMAKHWpnATQ4ChDoAQhKMAQ#v=onepage&q=runge kutta method is tvd proof&f=false starting from page 385 there is a equation: $$\mu_t = L(\mu) = -f(\mu)_x$$ and apply second or third order OPTIMAL TVD(total variation diminishing) Runge Kutta method to above equation. For instance towards third order OPTIMAL TVD(total variation diminishing) Runge Kutta method it is written that $$\mu^(1) = \mu^n + \Delta_t L(\mu^n)$$ $$\mu^(2) = 3/4\mu^n + 1/4\mu^(1) + 1/4\Delta_t L(\mu^(1))$$ $$\mu^(n+1) = 1/3\mu^n + 2/3mu^(2) + 2/3\Delta_t L(\mu^(2))$$ And this TVD Runge Kutta method is totally different from that form Runge Kutta for instance in http://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html
So my question is how do we derive the TVD Runge Kutta method?? How do we derive the coefficients such as 3/4 or 1/4 in the equation ##\mu^(2) = 3/4\mu^n + 1/4mu^(1) + 1/4\Delta_t L(\mu^(1))## ALSO there is optimal situation for that form of Runge Kutta, How do we derive the optimality in that form of TVD Runge Kutta??
So my question is how do we derive the TVD Runge Kutta method?? How do we derive the coefficients such as 3/4 or 1/4 in the equation ##\mu^(2) = 3/4\mu^n + 1/4mu^(1) + 1/4\Delta_t L(\mu^(1))## ALSO there is optimal situation for that form of Runge Kutta, How do we derive the optimality in that form of TVD Runge Kutta??