Narrow Results

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach.

The paper analyzes conditions for preserving the shape properties from the initial data to the limit curves of the binary three-point approximating subdivision scheme. We provide suitable conditions on the initial data utilizing the tension parameter $\omega$, thus the scheme can maintain three important shape properties, namely positivity, monotonicity and convexity in the limit curves. The use of derived conditions is illustrated in few examples, which offers more flexibility in the generation of smooth limit curves endowed with shape preserving properties.

A key difficulty of the conventional unstaggered central schemes for the shallow water equations (SWEs) is the well-balanced property that may be missed when the computational domain contains wet-dry fronts. To avoid the numerical difficulty caused by the nonconservative product, we construct a linear piecewise continuous bottom topography. We propose a new discretization of the source term on the staggered cells, and a novel “backward” step based on the water surface elevation. The core of this paper is that, we construct a map between the water surface elevation and the cell average of the free surface on the staggered cells to discretize the source term for maintaining the stationary solutions. The positivity-preserving property is obtained by using the “draining” time-step technique. A number of classical problems of the SWEs can be solved reasonably.

When one solves the Black–Scholes partial differential equation, it is of great important that numerical scheme to be free of spurious oscillations and satisfy the positivity requirement. With positivity, we mean, the component non-negativity of the initial vector, is preserved in time for the exact solution. Numerically, such property for fully implicit scheme is not always satisfied by approximated solutions and they generate spurious oscillations in the presence of discontinuous payoff. In this paper, by using the nonstandard discretization strategy, we propose a new scheme that is free of spurious oscillations and satisfies the positivity requirement.

Independent of the other communities who have developed theories of wavelets over the last twenty years, we developed over the same period a view of wavelets seen as stochastic processes. That context arose naturally from our theory of Time operators in statistical mechanics. Essential ingredients in our theory included Kolmogorov dynamical systems and conditional expectations. The purpose of the present paper is to come up-to-date on the relationship of our theory to the general theory of wavelets.