Criteria are developed for monotonicity of linear as well as nonlinear difference schemes associated with the numerical analysis of systems of partial differential equations, integrodifferential equations, etc. Difference schemes are converted into variational forms that satisfy the boundary maximum principle and also allow the investigation of monotonicity for nonlinear operators using linear ...

This paper concerns the theoretical analysis of step-by-step methods for solving initial value problems in ordinary and partial differential equations. The main theorem of the paper answers a natural question arising in the linear stability analysis of such methods. It guarantees a (strong) version of numerical stability – under a stepsize restriction related to the stability region of the nume...

The tail probabilities of two weighted sums of independent gamma random variables are compared when the first vector of weights majorizes the second vector of weights. The conjecture that the two cumulative distribution .functions cross exactly once is established in four special cases by means of the variation-diminishing property of totally positive kernels. Bounds are obtained for the locati...

We obtain upper bounds on the number of sign changes of linear combnatlons of derivatives and convolutions of Plya frequency functions using the variation diminishing properties of totally positive functions. These constitute extensions of earlier results of Karlin and Proschan.

This material supplements some sections of the paper entitled “Monte-Carlo SURE: A Black-Box Optimization of Regularization Parameters for General Denoising Algorithms”. Here, we elaborate on the solution to the differentiability issue associated with the Monte-Carlo divergence estimation proposed (in Theorem 2) in the paper. Firstly, we verify the validity of the Taylor expansion-based argumen...

We examine the application of a new implicit unconditionallystable high-resolution TVD scheme to steady-state calculations. It is a member of a one-parameter family of explicit and implicit second-order accurate schemes developed by Harten for the computation of weak solutions of one-dimensional hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a ...

In this paper, a new implicit nonstandard finite difference scheme for conservation laws, which preserving the property of TVD (total variation diminishing) of the solution, is proposed. This scheme is derived by using nonlocal approximation for nonlinear terms of partial differential equation. Schemes preserving the essential physical property of TVD are of great importance in practice. Such s...