Monotone Schemes Research Articles

Studies in Bayesian Econometrics and Statistics, In honor of Leonard J. Savage

We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.

On finite‐difference approximations and entropy conditions for shocks

AbstractWeak solutions of hyperbolic conservation laws are not uniquely determined by their initial values; an entropy condition is needed to pick out the physically relevant solution. The question arises whether finite‐difference approximations converge to this particular solution. It is shown in this paper that tin the case of a single conservation law, monotone schemes, when convergent, always converge to the physically relevant solution. Numerical examples show that this is not always the case with non‐monotone schemes, such as the Lax‐Wendroff scheme.

Fractional Brownian Motions, Fractional Noises and Applications

Fractional Brownian Motions, Fractional Noises and Applications