On Stability, Monotonicity, and Construction of Difference Schemes I: Theory

On stability of difference schemes. Central schemes for hyperbolic conservation laws with source terms

“…If ℵ = 0.75, then the subregion (42), called de Boor-Swartz box [27], coincides with the one used by Fritsch and Carlson [14]. It is proven in [7] that the interpolation (39) will be monotone, i.e. the value of an arbitrary component of p i+0.5 will be between the corresponding components of p i and p i+1 , iff (42) will be valid provided that 0 ≤ ℵ ≤ 1.…”

“…It can be done on the basis of the second order scheme (98), (99) with ease. Notice, adding to and subtracting from Equation (4.10) in [7] (rewritten for t n < t ≤ t n+1 ) the same quantity is equivalent to adding this quantity to (99) and subtracting it from (98). Let 0.125 (∆t) 2 (∂ 2 U ∂t 2 ) n+0.5 i be this quantity, then we obtain the following scheme, instead of (98), (99), v n+0.5…”

“…A new approach to testing scheme stability has been suggested in [7] to prove convergence of non-linear schemes for systems of PDEs. It was demonstrated [7] that the notion of scheme in variations (or variational scheme [9], [10]) has much potential to be an effective tool for studying stability of nonlinear schemes. The scheme in variations for a difference scheme represents a tangent space at a point of the manifold associated to the difference scheme [7], [9].…”

“…It was demonstrated [7] that the notion of scheme in variations (or variational scheme [9], [10]) has much potential to be an effective tool for studying stability of nonlinear schemes. The scheme in variations for a difference scheme represents a tangent space at a point of the manifold associated to the difference scheme [7], [9]. Thus, the scheme in variations will always be linear and, hence, enables the investigation of the stability for nonlinear operators using, in general, an infinite family of linear patterns.…”

“…In this paper, we extend the second-order modification to a new nonstaggered central scheme. The stability of this scheme is proven in Section 3.3 on the basis of the approach developed in Section 2 as well as in [7], [8]. The scheme is tested on several conservation laws in Section 5.…”