Total variation diminishing nonstandard finite difference schemes for conservation laws

Nonstandard finite difference schemes for conservation laws preserving the property of diminishing total variation of the solution are proposed. Computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step-size.