Nonnegativity Preservation Research Articles

Well-posedness for chemotaxis–fluid models in arbitrary dimensions* *The author acknowledges the support of the DAAD through the program ‘Graduate School Scholarship Programme, 2018’ (Number 57395813) and the Hausdorff Centre for Mathematics at Bonn.

We study the Cauchy problem for the chemotaxis Navier–Stokes equations and the Keller–Segel–Navier–Stokes system. Local-in-time and global-in-time solutions satisfying fundamental properties such as mass conservation and nonnegativity preservation are constructed for low regularity data in 2 and higher dimensions under suitable conditions. Our initial data classes involve a new scale of function space, that is which collects divergence of vector-fields with components in the square Campanato space , N > 2 (and can be identified with the homogeneous Besov space when N = 2) and are shown to be optimal in a certain sense. Moreover, uniqueness criterion for global solutions is obtained under certain limiting conditions.

A DEIM driven reduced basis method for the diffuse Stokes/Darcy model coupled at parametric phase-field interfaces

In this article, we develop a reduced basis method for efficiently solving the coupled Stokes/Darcy equations with parametric internal geometry. To accommodate possible changes in topology, we define the Stokes and Darcy domains implicitly via a phase-field indicator function. In our reduced order model, we approximate the parameter-dependent phase-field function with a discrete empirical interpolation method (DEIM) that enables affine decomposition of the associated linear and bilinear forms. In addition, we introduce a modification of DEIM that leads to non-negativity preserving approximations, thus guaranteeing positive-semidefiniteness of the system matrix. We also present a strategy for determining the required number of DEIM modes for a given number of reduced basis functions. We couple reduced basis functions on neighboring patches to enable the efficient simulation of large-scale problems that consist of repetitive subdomains. We apply our reduced basis framework to efficiently solve the inverse problem of characterizing the subsurface damage state of a complete in-situ leach mining site.

A nonnegativity preserving scheme for the relaxed Cahn–Hilliard equation with single-well potential and degenerate mobility

We propose and analyze a finite element approximation of the relaxed Cahn–Hilliard equation [Perthame and Poulain, Eur. J. Appl. Math. 32 (2021) 89–112.] with singular single-well potential of Lennard-Jones type and degenerate mobility that is energy stable and nonnegativity preserving. The Cahn–Hilliard model has recently been applied to model evolution and growth for living tissues. Although the choices of degenerate mobility and singular potential are biologically relevant, they induce difficulties regarding the design of a numerical scheme. We propose a finite element scheme, and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite element method. We propose two different time discretizations leading to a non-linear and a linear scheme. Moreover, we show the well-posedness and convergence of solutions of the non-linear numerical scheme. Finally, we validate our scheme by presenting numerical simulations in one and two dimensions.

A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford–Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate L1 space to a weak solution to the problem. By applying the methods and theory of operator semigroups, we are able to show that these weak solutions are unique and necessarily classical (differentiable) solutions, a degree of regularity not generally established when finite volume schemes are applied to such problems. Furthermore, this approach enabled us to derive a bound for the error induced by the truncation of the mass domain, and also establish the convergence of the truncated solutions as the truncation point is increased without bound. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence as the mesh is refined, whilst also verifying the bound on the truncation error.

A notion of non-negativity preserving relaxation for a mono-dimensional three velocities scheme with relative velocity

In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied.

High-accuracy numerical methods for a parabolic system in air pollution modeling

We present two approaches for enhancing the accuracy of the second-order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth-order compact difference scheme and the fourth-order scheme based on Richardson extrapolation. Our interest is concentrated on a system of ten parabolic partial differential equations in air pollution modeling. We analyze numerical experiments to compare the two approaches with respect to accuracy, computational complexity, nonnegativity preserving, etc. The sixth-order approximation based on the fourth-order compact difference scheme combined with Richardson extrapolation is also discussed numerically.

On the Markov commutator

The Markov commutator associated to a finite Markov kernel P is the convex semigroup consisting of all Markov kernels commuting with P. Its interest comes from its relation with the hypergroup property and with the notion of Markovian duality by intertwining. In particular, it is shown that the discrete analogue of the Achour-Trimèche's theorem, asserting the preservation of non-negativity by the wave equations associated to certain Metropolis birth and death transition kernels, cannot be extended to all convex potentials. But it remains true for symmetric and monotone potentials which are sufficiently convex.

Convergence and non-negativity preserving of the solution of balanced method for the delay CIR model with jump

In this work, we propose the balanced implicit method (BIM) to approximate the solution of the delay Cox–Ingersoll–Ross (CIR) model with jump which often gives rise to model an asset price and stochastic volatility dependent on past data. We show that this method preserves non-negativity property of the solution of this model with appropriate control functions. We prove the strong convergence and investigate the p th moment boundedness of the solution of BIM. Finally, we illustrate those results in the last section.

Cauchy problems for Keller–Segel type time–space fractional diffusion equation

This paper investigates Cauchy problems for nonlinear fractional time–space generalized Keller–Segel equation Dtβ0cρ+(−△)α2ρ+∇⋅(ρB(ρ))=0, where Caputo derivative Dtβ0cρ models memory effects in time, fractional Laplacian (−△)α2ρ represents Lévy diffusion and B(ρ)=−sn,γ∫Rnx−y|x−y|n−γ+2ρ(y)dy is the Riesz potential with a singular kernel which takes into account the long rang interaction. We first establish Lr−Lq estimates and weighted estimates of the fundamental solutions (P(x,t),Y(x,t)) (or equivalently, the solution operators (Sαβ(t),Tαβ(t))). Then, we prove the existence and uniqueness of the mild solutions when initial data are in Lp spaces, or the weighted spaces. Similar to Keller–Segel equations, if the initial data are small in critical space Lpc(Rn) (pc=nα+γ−2), we construct the global existence. Furthermore, we prove the L1 integrability and integral preservation when the initial data are in L1(Rn)∩Lp(Rn) or L1(Rn)∩Lpc(Rn). Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established.

Non-Negativity Preserving Numerical Algorithms for Problems in Mathematical Finance

We give a study result to analyze a rather different, semi-analytical numerical algorithms based on splitting-step methods with their applications to mathematical finance. As certain subsistent numerical schemes may fail due to producing negative values for financial variables which require non-negativity preserving. These algorithms which we are analyzing preserve not only the non-negativity, but also the character of boundaries (natural, reflecting, absorbing, etc.). The derivatives of the CIR process and the Heston model are being extensively studied. Beyond plain vanilla European options, we creatively apply our splitting-step methods to a path-dependent option valuation. We compare our algorithms to a class of numerical schemes based on Euler discretization which are prevalent currently. The comparisons are given with respect to both accuracy and computational time for the European call option under the CIR model whereas with respect to convergence rate for the path-dependent option under the CIR model and the European call option under the Heston model.

On Generalized Proportional Allocation Policies for Traffic Signal Control

The fast-increasing demand and relatively slow growth of infrastructure capacity are providing a strong motivation for research in real-time urban traffic controls that make the best use of novel sensing in order to increase efficiency and resilience of the transportation system. In our contribution, we focus on a class of dynamic feedback traffic signal control policies that are based on a generalized proportional allocation rule. The proposed traffic signal controls are decentralized (they make use of local information only), scalable (they are independent of the network size and topology), and universal (they do not rely on any information about external inflows or turning ratios). In spite of their fully distributed nature, we prove that such control policies achieve a global objective, maximum throughput, in that they stabilize the urban traffic network whenever possible under the given capacity constraints.The traffic model we consider consists in a network of interconnected vertical queues with deterministic dynamics driven by physical laws (conservation of mass and preservation of non-negativity of the traffic volumes) as well as scheduling constraints (described as a set of phases, each phase consisting in a subset of lanes that can be be given green light simultaneously). This results in a differential inclusion for which we prove existence and, in the special case of orthogonal phases, uniqueness of continuous solutions via a generalization of the reflection principle. Stability is then proved by interpreting the generalized proportional allocation controllers as minimizers of a certain entropy-like function that is then used as a Lyapunov function for the closed-loop system.

A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations

Preservation of nonnengativity and boundedness in the finite element solution of Nagumo-type equations with general anisotropic diffusion is studied. Linear finite elements and the backward Euler scheme are used for the spatial and temporal discretization, respectively. An explicit, an implicit, and two hybrid explicit–implicit treatments for the nonlinear reaction term are considered. Conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness. The effects of lumping of the mass matrix and the reaction term are also discussed. The analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size. Numerical examples are given to demonstrate the theoretical findings.

A well-balanced scheme for the shallow-water equations with topography or Manning friction

We consider the shallow-water equations with Manning friction or topography, as well as a combination of both these source terms. The main purpose of this work concerns the derivation of a non-negativity preserving and well-balanced scheme that approximates solutions of the system and preserves the associated steady states, including the moving ones. In addition, the scheme has to deal with vanishing water heights and transitions between wet and dry areas. To address such issues, a particular attention is paid to the study of the steady states related to the friction source term. Then, a Godunov-type scheme is obtained by using a relevant average of the source terms in order to enforce the required well-balance property. An implicit treatment of both topography and friction source terms is also exhibited to improve the scheme while dealing with vanishing water heights. A second-order well-balanced MUSCL extension is designed, as well as an extension for the two-dimensional case. Numerical experiments are performed in order to highlight the properties of the scheme.

On nonnegativity preservation in finite element methods for subdiffusion equations

We consider three types of subdiffusion models, namely singleterm, multi-term and distributed order fractional diffusion equations, for which the maximum-principle holds and which, in particular, preserve nonnegativity. Hence the solution is nonnegative for nonnegative initial data. Following earlier work on the heat equation, our purpose is to study whether this property is inherited by certain spatially semidiscrete and fully discrete piecewise linear finite element methods, including the standard Galerkin method, the lumped mass method and the finite volume element method. It is shown that, as for the heat equation, when the mass matrix is nondiagonal, nonnegativity is not preserved for small time or time-step, but may reappear after a positivity threshold. For the lumped mass method nonnegativity is preserved if and only if the triangulation in the finite element space is of Delaunay type. Numerical experiments illustrate and complement the theoretical results.

Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations

Two fully discrete methods are investigated for simulating the distributed-order sub-diffusion equation in Caputo’s form. The fractional Caputo derivative is approximated by the Caputo’s BDF1 (called L1 early) and BDF2 (or L1-2 when it was first introduced) approximations, which are constructed by piecewise linear and quadratic interpolating polynomials, respectively. It is shown that the first scheme, using the BDF1 formula, possesses the discrete minimum-maximum principle and nonnegativity preservation property such that it is stable and convergent in the maximum norm. The method using the BDF2 formula is shown to be stable and convergent in the discrete H1 norm by using the discrete energy method. For problems of distributed order within a certain region, the method is also proven to preserve the discrete maximum principle and nonnegativity property. Extensive numerical experiments are provided to show the effectiveness of numerical schemes, and to examine the initial singularity of the solution. The applicability of our numerical algorithms to a problem with solution which lacks the smoothness near the initial time is examined by employing a class of power-type nonuniform meshes.

A well-balanced scheme for the shallow-water equations with topography

A non-negativity preserving and well-balanced scheme that exactly preserves all the smooth steady states of the shallow water system, including the moving ones, is proposed. In addition, the scheme must deal with vanishing water heights and transitions between wet and dry areas. A Godunov-type method is derived by using a relevant average of the source terms within the scheme, in order to enforce the required well-balance property. A second-order well-balanced MUSCL extension is also designed. Numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states.

Energy Stable Model Reduction of Neurons by Nonnegative Discrete Empirical Interpolation

The accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences. This work develops a novel structure-preserving model reduction technique to address this problem based on Galerkin projection and nonnegative operator approximation. It is first shown that the corresponding reduced-order model is guaranteed to be energy stable, thanks to both the structure-preserving approach that constructs a distinct reduced-order basis for each cable in the network and the preservation of nonnegativity. Furthermore, a posteriori error estimates are provided, showing that the model reduction error can be bounded and controlled. Finally, the application to the model reduction of a large-scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups.

Scattered data interpolation with nonnegative preservation using bivariate splines and its application

We study how to use bivariate splines for scattered data interpolation with nonnegativity preservation of the given data values. That is, we propose a constrained minimal energy method to find a C1 smooth interpolation of nonnegative data values over scattered locations using bivariate splines. We establish the existence and uniqueness of the minimizer under mild assumptions on the data locations and triangulations. Next we show some approximation properties of the minimizer. We then use the classic projected gradient algorithm to find the minimizer using a simplified nonnegative constraint. Some synthetic as well as a real life example demonstrate the performance, producing nonnegative interpolatory spline surfaces from nonnegative data values. Experimental results also show that the approximation of the nonnegative interpolatory spline surfaces is slightly better than the approximation of the classic minimal energy interpolatory splines.

An explicit residual based approach for shallow water flows

We describe fully explicit residual based discretizations of the shallow water equations with friction on unstructured grids. The schemes are obtained by properly adapting the explicit construction proposed in Ricchiuto and Abgrall (2010) [57]. In particular, previous work on well balanced integration (Ricchiuto, 2011 [56]) and preservation of the depth non-negativity (Ricchiuto and Bollermann, 2009 [60]) is reformulated in the context of a genuinely explicit time stepping still based on a weighted residual approximation. The paper discusses in depth how to achieve in this context an exact preservation of all the simple known steady equilibria, and how to obtain a super-consistent approximation for smooth non-trivial moving equilibria. The treatment of the wetting/drying interface is also discussed, giving formal conditions for the preservation of the non-negativity of the depth for a particular case, based on a nonlinear variant of a Lax–Friedrichs type scheme. The approach is analyzed and tested thoroughly. The quality of the numerical results shows the interest in the proposed approach over previously proposed schemes, in terms of accuracy and efficiency.

Nonnegativity Preserving Interpolation by C1 Bivariate Rational Spline Surface

This paper is concerned with the nonnegativity preserving interpolation of data on rectangular grids. The function is a kind of bivariate rational interpolation spline with parameters, which is C1 in the whole interpolation region. Sufficient conditions are derived on coefficients in the rational spline to ensure that the surfaces are always nonnegative if the original data are nonnegative. The gradients at the data sites are modified if necessary to ensure that the nonnegativity conditions are fulfilled. Some numerical examples are illustrated in the end of this paper.