A boundedness-preserving finite-difference scheme for a damped nonlinear wave equation

On some explicit non-standard methods to approximate nonnegative solutions of a weakly hyperbolic equation with logistic nonlinearity

“…A simple consistency analysis establishes in this case that finitedifference schemes (16), (17), (18) and (19) are, in general, consistent approximations to the solutions of (2) of order O( t + ( x) 2 ). Indeed, it is readily observed that the standard differences satisfy the properties…”

“…In order to find conditions under which the finite-difference schemes presented in the previous subsection be nonnegative, it must be noticed beforehand that the coefficients of the term u k n+1 + u k n−1 in all four schemes are Table 1. Expressions of the four parametric functions k 1 , k 2 , k 3 and k 4 of u k n , for each of the Schemes (16), (17), (18) and (19) when presented in the explicit form (20).…”

“…In the former case, the conditions assume the discrete form u k 0 = a and Figure 1. Forward-difference stencil for the approximation to the partial differential equation (2) at time t k , for R = 1 4 , using Schemes (16), (17) and (18) with α given by Equation (25) and Scheme (19) with α provided by Equation (28). The black circles represent known approximations to the actual solutions at times t k−1 and t k in the schemes, and the crosses denote the unknown approximation at time t k+1 .…”

“…The methods presented here have three levels, are explicit, and the numerical results evidence an excellent agreement between the approximations provided by the methods and the exact solutions employed here. Moreover, they represent a computational advantage with respect to a previous work by these authors [18]. Section 2 introduces the equation under investigation in this work, together with the exact type of nonlinearity considered.…”

A dynamical study of the evolution of pressure waves propagating through a semi-infinite region of homogeneous gas combustion subject to a time-harmonic signal at the boundary

“…In Chapter 6, the perturbation method is compared with a positivity and boundedness preserving non-standard finite-difference scheme [18,22]. First, convergence of the finite-difference solution ρ is established for one quarter wavelength by computing ρ at one spatial-step to the left of the right boundary for several decreasing mesh sizes, ( ) , t x ∆ ∆ .…”