TVD (total variable diminishing) and TVB methods for solving initial value problems I trying to find out what TVD (total variable diminishing) and TVB methods are and I want to compare them with runge kutta method. the stuff I have already found are seem complex for me.

I learned TVD and TVB methods for scalar conservation laws from Hyperbolic systems of conservation laws, by E. Godlewski and P-A. Raviart. It is quite an old book already, but it explains quite nicely how these type of methods arise.

Another book I used to read was Numerical Methods for Conservation Laws by R. J. LeVeque. I think there ares several editions of this book; you can find an old (but enough) edition here.

Unfortunatelly, I do not remember if there is anything about Runge-Kutta methods.