Monotonicity Preserving Research Articles

Two boundedness and monotonicity preserving methods for a generalized Fisher-KPP equation

A semi-explicit finite difference method and an implicit finite difference method are proposed for a generalized Fisher-KPP equation with two space variables. It is proved that these two methods could preserve the skew-symmetry of this equation. Moreover, under a condition on the step sizes, the semi-explicit method is capable of preserving the positivity, the boundedness, and the spatial and temporal monotonicity of initial approximations. And the implicit method is able to preserve these properties with no restriction on the step sizes. The stability and convergence of these two methods are also analyzed respectively. Finally, some numerical simulations are provided to verify the validity of our analytical results.

Extended Brauer model for ferromagnetic materials: analysis and computation

Purpose – The purpose of this paper is to develop a fast and accurate analytic model function for the single-valued H-B curve of ferromagnetic materials, where hysteresis can be disregarded (normal magnetization curve). Nonlinear magnetoquasistatic simulations demand smooth monotone material models to ensure physical correctness and good convergence in Newton's method. Design/methodology/approach – The Brauer model has these beneficial properties, but is not sufficiently accurate for low and high fields in the normal magnetization curve. The paper extends the Brauer model to better fit material behavior in the Rayleigh region (low fields) and in full saturation. Procedures for obtaining optimal parameters from given measurement points are proposed and tested for two technical materials. The approach is compared with cubic spline and monotonicity preserving spline interpolation with respect to error and computational effort. Findings – The extended Brauer model is more accurate and even maintains the computational advantages of the classical Brauer model. The methods for obtaining optimal parameters yield good results if the measurement points have a distinctive Rayleigh region. Originality/value – The model function for ferromagnetic materials enhances the precision of the classical Brauer model without notable additional simulation cost.

On splitting-based numerical methods for nonlinear models of European options

We present a large class of nonlinear models of European options as parabolic equations with quasi-linear diffusion and fully nonlinear hyperbolic part. The main idea of the operator splitting method (OSM) is to couple known difference schemes for nonlinear hyperbolic equations with other ones for quasi-linear parabolic equations. We use flux limiter techniques, explicit–implicit difference schemes, Richardson extrapolation, etc. Theoretical analysis for illiquid market model is given. The numerical experiments show second-order accuracy for the numerical solution (the price) and Greeks Delta and Gamma, positivity and monotonicity preserving properties of the approximations.

Shape Preserving Properties forq-Bernstein-Stancu Operators

We investigate shape preserving forq-Bernstein-Stancu polynomialsBnq,α(f;x)introduced by Nowak in 2009. Whenα=0,Bnq,α(f;x)reduces to the well-knownq-Bernstein polynomials introduced by Phillips in 1997; whenq=1,Bnq,α(f;x)reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; whenq=1,α=0, we obtain classical Bernstein polynomials. We prove that basicBnq,α(f;x)basis is a normalized totally positive basis on[0,1]andq-Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on[0,1].

A new four-point shape-preserving [formula omitted] subdivision scheme

A new binary four-point subdivision scheme is presented, which keeps the second-order divided difference at the old vertices unchanged when the new vertices are inserted. Using the symbol of the subdivision scheme, we show that the limit curve is at least C3 continuous. Furthermore, the conditions imposed on the initial points are also discussed, thus the given limit functions are both monotonicity preserving and convexity preserving

Weighted Rational Quartic Spline Interpolation

The weighted rational quartic spline interpolation has been constructed using two kinds of rational quartic spline with linear denominator in this paper. And the conditions of monotonicity preserving and C 2 continuity have been discussed. In addition to, the problem that constrains the weighted interpolating curves to lie above or below a given straight line or quadratic curve has been solved.

A Remapping Method Based on Multi-Point Flux Corner Transport Upwind Advection Algorithm

A local remapping algorithm for scalar function on quadrilateral meshes is described. The remapper from a distorted grid to a rezoned grid is usually regarded as a conservative interpolation problem. The present paper introduces a pseudo time to transform the interpolation into an initial value problem on a moving grid, and construct a moving mesh method to solve it. The new feature of the algorithm is the introduction of multipoint information on each edge, which leads to the numerical flux consistent with grid node motion. During the procedure of deriving scheme, we illustrate a framework about how the algorithms on a rectangular mesh are easily generated to those on a moving mesh. The basic ideas include: (i) introducing coordinate transformation, which maps the irregular domain in physical space to a perfectly regular computational domain, and (ii) deriving finite volume methods in the physical domain, which can be viewed as a discretization of the transformed equation. The resulting scheme is second-order accurate, conservative and monotonicity preserving. Numerical examples are carried out to show the good performance of our schemes.

3DFLUX: A high-order fully three-dimensional flux integral solver for the scalar transport equation

We present a detailed derivation of a high-order, fully three-dimensional, conservative, monotonicity preserving, flux integral method for the solution of the scalar transport equation. This algorithm, named 3DFLUX, produces highly accurate solutions that are nearly unaffected by numerical dissipation, at a realistic computational cost. The performance of 3DFLUX is characterized by means of several challenging multidimensional tests. 3DFLUX is nominally third-order in space and second-order in time, however, at low Courant numbers, it appears to be superconvergent and, depending on the problem solved, is fourth-order or higher in space. Finally, 3DFLUX is used to simulate advection-diffusion of a complex temperature field in an incompressible turbulent flow of practical relevance, and its results are in excellent agreement with experimental measurements.

AUSM-Based High-Order Solution for Euler Equations

AbstractIn this paper we demonstrate the accuracy and robustness of combining the advection upwind splitting method (AUSM), specifically AUSM+-UP, with high-order upwind-biased interpolation procedures, the weighted essentially non-oscillatory (WENO-JS) scheme and its variations, and the monotonicity preserving (MP) scheme, for solving the Euler equations. MP is found to be more effective than the three WENO variations studied. AUSM+-UP is also shown to be free of the so-called “carbuncle” phenomenon with the high-order interpolation. The characteristic variables are preferred for interpolation after comparing the results using primitive and conservative variables, even though they require additional matrix-vector operations. Results using the Roe flux with an entropy fix and the Lax-Friedrichs approximate Riemann solvers are also included for comparison. In addition, four reflective boundary condition implementations are compared for their effects on residual convergence and solution accuracy. Finally, a measure for quantifying the efficiency of obtaining high order solutions is proposed; the measure reveals that a maximum return is reached after which no improvement in accuracy is possible for a given grid size.

A high‐order finite difference method for numerical simulations of supersonic turbulent flows

AbstractThis paper describes a new class of three‐dimensional finite difference schemes for high‐speed turbulent flows in complex geometries based on the high‐order monotonicity‐preserving (MP) method. Simulations conducted for various 1D, 2D, and 3D problems indicate that the new high‐order MP schemes can preserve sharp changes in the flow variables without spurious oscillations and are able to capture the turbulence at the smallest computed scales. Our results also indicate that the MP method has less numerical dissipation and faster grid convergence than the weighted essentially non‐oscillatory method. However, both of these methods are computationally more demanding than the COMP method and are only used for the inviscid fluxes. To reduce the computational cost for reacting flows, the scalar equations are solved by the COMP method, which is shown to yield similar results to those obtained by the MP in supersonic turbulent flows with strong shock waves. Copyright © 2011 John Wiley & Sons, Ltd.

High-order conservative finite difference GLM–MHD schemes for cell-centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. [J. Comput. Phys. 175 (2002) 645–673]. The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.

Monotonicity preserving representations of non-polynomial surfaces

We prove that, in contrast to the case for rational surfaces, some tensor product representations through spaces containing algebraic, trigonometric and hyperbolic polynomials are monotonicity preserving. The surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product Bézier and tensor product B-spline surfaces.

Selective monotonicity preservation in scalar advection

An efficient method for scalar advection is developed that selectively preserves monotonicity. Monotonicity preservation is applied only where the scalar field is likely to contain discontinuities as indicated by significant grid-cell-to-grid-cell variations in a smoothness measure conceptually similar to that used in weighted essentially non-oscillatory (WENO) methods. In smooth regions, the numerical diffusion associated with monotonicity-preserving methods is avoided. The resulting method, while not globally monotonicity preserving, allows the full accuracy of the underlying advection scheme to be achieved in smooth regions. The violations of monotonicity that do occur are generally very small, as seen in the tests presented here. Strict positivity preservation may be effectively and efficiently obtained through an additional flux correction step. The underlying advection scheme used to test this methodology is a variant of the piecewise parabolic method (PPM) that may be applied to multi-dimensional problems using density-corrected dimensional splitting and permits stable semi-Lagrangian integrations using CFL numbers larger than one. Two methods for monotonicity preservation are used here: flux correction and modification of the underlying polynomial reconstruction.

A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes

Runge–Kutta Discontinuous Galerkin (RKDG) schemes can provide highly accurate solutions for a large class of important scientific problems. Using them for problems with shocks and other discontinuities requires that one has a strategy for detecting the presence of these discontinuities. Strategies that are based on total variation diminishing (TVD) limiters can be problem-independent and scale-free but they can indiscriminately clip extrema, resulting in degraded accuracy. Those based on total variation bounded (TVB) limiters are neither problem-independent nor scale-free. In order to get past these limitations we realize that the solution in RKDG schemes can carry meaningful sub-structure within a zone that may not need to be limited. To make this sub-structure visible, we take a sub-cell approach to detecting zones with discontinuities, known as troubled zones. A monotonicity preserving (MP) strategy is applied to distinguish between meaningful sub-structure and shocks. The strategy does not indiscriminately clip extrema and is, nevertheless, scale-free and problem-independent. It, therefore, overcomes some of the limitations of previously-used strategies for detecting troubled zones. The moments of the troubled zones can then be corrected using a weighted essentially non-oscillatory (WENO) or Hermite WENO (HWENO) approach. In the course of doing this work it was also realized that the most significant variation in the solution is contained in the solution variables and their first moments. Thus the additional moments can be reconstructed using the variables and their first moments, resulting in a very substantial savings in computer memory. We call such schemes hybrid RKDG+HWENO schemes. It is shown that such schemes can attain the same formal accuracy as RKDG schemes, making them attractive, low-storage alternatives to RKDG schemes. Particular attention has been paid to the reconstruction of cross-terms in multi-dimensional problems and explicit, easy to implement formulae have been catalogued for third and fourth order of spatial accuracy. The utility of hybrid RKDG+WENO schemes has been illustrated with several stringent test problems in one and two dimensions. It is shown that their accuracy is usually competitive with the accuracy of RKDG schemes of the same order. Because of their compact stencils and low storage, hybrid RKDG+HWENO schemes could be very useful for large-scale parallel adaptive mesh refinement calculations.

Are rational Bézier surfaces monotonicity preserving?

We present examples which show that rational Bézier surfaces are not axially monotonicity preserving and that surfaces generated by the tensor product of rational bases are not monotonicity preserving. Besides, we prove that surfaces generated by rational functions through Bernstein basis on a triangle are not axially monotonicity preserving.

Monotonicity preserving multigrid time stepping schemes for conservation laws

In this paper, we propose monotonicity preserving and total variation diminishing (TVD) multigrid methods for solving scalar conservation laws. We generalize the upwind-biased residual restriction and interpolation operators for solving linear wave equations to nonlinear conservation laws. The idea is to define nonlinear restriction and interpolation based on local Riemann solutions. Theoretical analyses have been provided to analyze the monotonicity preserving and TVD properties of the resulting multigrid time stepping schemes. Numerical results are given to verify the theoretical results and demonstrate the effectiveness of the proposed schemes. Two dimensional extension is also discussed.

Diffusion-Inspired Shrinkage Functions and Stability Results for Wavelet Denoising

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of a (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet denoising. Moreover, by transferring stability notions from diffusion filtering to wavelet shrinkage, we derive conditions on the shrinkage function that ensure that shift invariant single-level Haar wavelet shrinkage is maximum---minimum stable, monotonicity preserving, and variation diminishing.

Corner cutting systems

If we can evaluate a curve by means of a corner cutting algorithm, we call the corresponding system of functions a corner cutting system. We prove that corner cutting systems are always monotonicity preserving. We obtain weak sufficient conditions for corner cutting systems. Optimal stability properties and properties concerning the supports of the basis functions of corner cutting systems are also analyzed.

On Monotonicity of Difference Schemes for Computational Physics

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 patterns. Sufficient conditions are provided to review the monotonicity of single and coupled difference schemes. Necessary as well as necessary and sufficient conditions for monotonicity of explicit schemes are also developed. The notion of submonotone difference schemes is considered and the associated criteria are developed. We discuss the interrelationship between monotonicity, submonotonicity, and stability. Some known schemes serve as examples demonstrating the implementation of the developed approaches. Among these examples, we describe the possibility that stable schemes such as total variation diminishing (TVD) as well as monotonicity preserving can produce spurious oscillations.

High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations

This paper deals with the development of accurate one-step schemes for the numerical simulation of unsteady compressible flows. Pursuing our work in Daru and Tenaud [V. Daru, C. Tenaud, Comput. Fluids 30 (2001) 89] where third-order schemes were considered, we follow the Lax–Wendroff approach to develop high order TVD combined time–space schemes by correcting the successive modified equations. In the scalar case, TVD schemes accurate up to seventh order (OSTVD7) in time and space are obtained (in smooth regions and away from extrema). To avoid the clipping and the loss of accuracy that is common to the TVD schemes near extrema, we develop monotonicity-preserving (MP) conditions derived from Suresh and Huynh [A. Suresh, H.T. Huynh, J. Comput. Phys. 136 (1997) 83] to locally relax the TVD limitation for this family of one-step schemes. Numerical results for long time integration in the scalar case show that the MP one-step approach gives the best results compared to several multistage schemes, including WENO schemes. The extension to systems and to the multidimensional case is done in a simplified way which does not preserve the scalar order of accuracy. However we show that the resulting schemes have a very low level of error. For validation, the present algorithm has been checked on several classical one-dimensional and multidimensional test cases, including both viscous and inviscid flows: a moving shock wave interacting with a sine wave, the Lax shock tube problem, the 2D inviscid double Mach reflection and the 2D viscous shock wave–vortex interaction. By computing these various test cases, we demonstrate that very accurate results can be obtained by using the one-step MP approach which is very competitive compared to multistage high order schemes.