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High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles. [ABSTRACT FROM AUTHOR]

The equations of Lagrangian gas dynamics fall into the larger class of overdetermined hyperbolic and thermodynamically compatible (HTC) systems of partial differential equations. They satisfy an entropy inequality (second principle of thermodynamics) and conserve total energy (first principle of thermodynamics). The aim of this work is to construct a novel thermodynamically compatible cell-centered Lagrangian finite volume scheme on unstructured meshes. Unlike in existing schemes, we choose to directly discretize the entropy inequality, hence obtaining total energy conservation as a consequence of the new thermodynamically compatible discretization of the other equations. First, the governing equations are written in fluctuation form. Next, the noncompatible centered numerical fluxes are corrected according to the approach recently introduced by Abgrall et al. using a scalar correction factor that is defined at the nodes of the grid. This perfectly fits into the formalism of nodal solvers which is typically adopted in cell-centered Lagrangian finite volume methods. Semidiscrete entropy conservative and entropy stable Lagrangian schemes are devised, and they are adequately blended together via a convex combination based on either a priori or a posteriori detectors of discontinuous solutions. The nonlinear stability in the energy norm is rigorously demonstrated, and the new schemes are provably positivity preserving for density and pressure. Furthermore, they exhibit zero numerical diffusion for isentropic flows while still being nonlinearly stable. The new schemes are tested against classical benchmarks for Lagrangian hydrodynamics, assessing their convergence and robustness and comparing their numerical dissipation with classical Lagrangian finite volume methods. [ABSTRACT FROM AUTHOR]

We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the L² projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient L² projection part), positivity preserving, and mass conserving. Rigorous error estimates in the L² norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., H¹ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We develop the numerical analysis of an energy-dissipating finite volume scheme on admissible meshes for the non-local, nonlinear aggregation-diffusion equations. Crucially, this scheme keeps the dissipation property unconditionally of an associated fully discrete energy, and preserves the mass and positivity conservation of the density. We establish the well-posedness, stability and error analysis of the method. Several numerical examples are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In 2014, Neuenkirch and Szpruch established the drift-implicit Euler-Maruyama method for a class of SDEs which take values in a given domain. However, expensive computational cost is required for implementation of an implicit numerical method. A competitive positivity preserving explicit numerical method for SDEs which take values in the positive domain is the logarithmic truncated Euler-Maruyama method. However, assumptions for the logarithmic truncated Euler-Maruyama method used in previous work are restrictive which exclude some important SDE models with specific parameters. The main aim of this paper is to use weaker assumptions to establish strong convergence theory for the logarithmic truncated Euler-Maruyama method. [ABSTRACT FROM AUTHOR]

In this study, we propose positivity preserving conservative characteristic methods with temporal second-order and spatial fourth-order accuracy for solving convection dominated diffusion problems. The method of characteristics is utilized to avoid strict restrictions on time step sizes, providing greater flexibility in computation. To preserve mass, conservative piecewise parabolic interpolation is used to obtain values at tracking points. Additionally, we leverage the finite difference implementation of the continuous finite element method and construct various fourth-order approximation operators for the Laplace operator, which are then applied to develop conservative numerical schemes with positivity preserving property. The proposed methods are theoretically proven to preserve the positivity property of solutions and ensure mass conservation. Numerical examples are conducted to validate the performance of developed schemes, demonstrating their spatial and temporal convergence orders, as well as conservation property. [ABSTRACT FROM AUTHOR]

In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. A stability investigation for these methods, which are outside the class of general linear methods, is challenging since the iterates are always generated by a nonlinear map even for linear problems. Recently, a stability theorem was derived presenting criteria for understanding such schemes. For the analysis, the schemes are applied to general linear equations and proven to be generated by C 1 -maps with locally Lipschitz continuous first derivatives. As a result, the above mentioned stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it. In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge–Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically. [ABSTRACT FROM AUTHOR]

A second order accurate (in time) numerical scheme is proposed and analyzed for the Poisson–Nernst–Planck equation (PNP) system, reformulated as a non-constant mobility H - 1 gradient flow in the Energetic Variational Approach (EnVarA). The centered finite difference is taken as the spatial discretization. Meanwhile, the highly nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty to design a second order accurate scheme in time, while preserving the variational energetic structures. The mobility function is updated with a second order accurate extrapolation formula, for the sake of unique solvability. A modified Crank–Nicolson scheme is used to approximate the logarithmic term, so that its inner product with the discrete temporal derivative exactly gives the corresponding nonlinear energy difference; henceforth the energy stability is ensured for the logarithmic part. In addition, nonlinear artificial regularization terms are added in the numerical scheme, so that the positivity-preserving property could be theoretically proved, with the help of the singularity associated with the logarithmic function. Furthermore, an optimal rate convergence analysis is provided in this paper, in which the higher order asymptotic expansion for the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This work combines the following theoretical properties for a second order accurate numerical scheme for the PNP system: (i) second order accuracy in both time and space, (ii) unique solvability and positivity, (iii) energy stability, and (iv) optimal rate convergence. A few numerical results are also presented. [ABSTRACT FROM AUTHOR]

The well-suited discretization of the Keller–Segel equations for chemotaxis has become a very challenging problem due to the convective nature inherent to them. This paper aims to introduce a new upwind, mass-conservative, positive and energy-dissipative discontinuous Galerkin scheme for the Keller–Segel model. This approach is based on the gradient-flow structure of the equations. In addition, we show some numerical experiments in accordance with the aforementioned properties of the discretization. The numerical results obtained emphasize the really good behaviour of the approximation in the case of chemotactic collapse, where very steep gradients appear. [ABSTRACT FROM AUTHOR]

We propose and analyze a first-order in time, second order in space finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Huggins potential. The semi-implicit numerical scheme is designed based on a suitable convex-concave decomposition of the FCH free energy. We prove unique solvability of the numerical algorithm and verify its unconditional energy stability without any restriction on the time step size. Thanks to the singular nature of the logarithmic part in the Flory-Huggins potential near the pure states ± 1 , we establish the so-called positivity-preserving property for the phase function at a theoretic level. As a consequence, the numerical solutions will never reach the singular values ± 1 in the point-wise sense and the fully discrete scheme is well defined at each time step. Next, we present a detailed optimal rate convergence analysis and derive error estimates in l ∞ (0 , T ; L h 2) ∩ l 2 (0 , T ; H h 3) under a linear refinement requirement Δ t ≤ C 1 h . To achieve the goal, a higher order asymptotic expansion (up to the second order temporal and spatial accuracy) based on the Fourier projection is utilized to control the discrete maximum norm of solutions to the numerical scheme. We show that if the exact solution to the continuous problem is strictly separated from the pure states ± 1 , then the numerical solutions can be kept away from ± 1 by a positive distance that is uniform with respect to the size of the time step and the grid. Finally, a few numerical experiments are presented. Convergence test is performed to demonstrate the accuracy and robustness of the proposed numerical scheme. Pearling bifurcation, meandering instability and spinodal decomposition are observed in the numerical simulations. [ABSTRACT FROM AUTHOR]

This work focuses on the robust and high-order numerical simulations of weakly compressible two-phase flows by using the discontinuous Galerkin (DG) method combined with an explicit strong-stability-preserving Runge–Kutta scheme. In order to improve the computational robustness under large density ratios, a nonlinear weighted essentially non-oscillatory (WENO) limiter and a positivity-preserving limiter are specially designed and applied with the aim of dampening the nonphysical oscillations around the phase interface and preventing the occurrence of negative density, respectively. More importantly, we theoretically prove that the present method is able to satisfy the uniform-pressure–velocity criterion which states that uniform pressure and velocity profiles around an isolated phase interface should be preserved during the simulation. The performance of the present method is validated by a range of benchmark test cases with density ratios up to 1000:1. The results demonstrate that the present method possesses a good capability of simulating weakly compressible two-phase flows with large density ratios. [ABSTRACT FROM AUTHOR]

Two standard and two nonstandard finite difference schemes are constructed to solve a basic reaction–diffusion–chemotaxis model, for which no exact solution is known. The continuous model involves a system of nonlinear coupled partial differential equations subject to some specified initial and boundary conditions. It is not possible to obtain theoretically the stability region of the two standard finite difference schemes. Through running some numerical experiments, we deduce heuristically that these classical methods give reasonable solutions when the temporal step size kk is chosen such that k≤0.25k\le 0.25 with the spatial step size hh fixed at h=1.0h=1.0 (first novelty of this work). We observe that the standard finite difference schemes are not always positivity preserving, and this is why we consider nonstandard finite difference schemes. Two nonstandard methods abbreviated as NSFD1 and NSFD2 from Chapwanya et al. are considered. NSFD1 was not used by Chapwanya et al. to generate results for the basic reaction–diffusion–chemotaxis model. We find that NSFD1 preserves positivity of the continuous model if some criteria are satisfied, namely, ϕ(k)[ψ(h)]2=12γ≤12σ+β\frac{\phi \left(k)}{{\left[\psi \left(h)]}^{2}}=\frac{1}{2\gamma }\le \frac{1}{2\sigma +\beta } and β≤σ\beta \le \sigma , and this is the second novelty of this work. Chapwanya et al. modified NSFD1 to obtain NSFD2, which is positivity preserving if R=ϕ(k)[ψ(h)]2=12γR=\frac{\phi \left(k)}{{\left[\psi \left(h)]}^{2}}=\frac{1}{2\gamma } and 2σR≤12\sigma R\le 1, that is σ≤γ\sigma \le \gamma , and they presented some results. For the third highlight of this work, we show that NSFD2 is not always consistent and prove that consistency can be achieved if β→0\beta \to 0 and kh2→0\frac{k}{{h}^{2}}\to 0. Fourthly, we show numerically that the rate of convergence in time of the four methods for case 2 is approximately one.

A large class of differential Riccati equations (DREs) satisfy positivity property in the sense that the time-dependent solution preserves for any time its symmetric and positive semidefinite structure. This positivity property plays a crucial role in understanding the wellposedness of the DRE, and whether it could be inherited in the discrete level is a significant issue in numerical simulations. In this paper, we study positivity preserving time integration schemes by means of exponential integrators. The proposed exponential Euler and exponential midpoint schemes are linear and proven to be positivity preserving and unconditionally stable. Sharp error estimates of the schemes are also obtained. Numerical experiments are carried out to illustrate the performance of the proposed integrators. [ABSTRACT FROM AUTHOR]

In this study, a numerical solution for degenerate space–time fractional advection–dispersion equations is proposed to simulate atmospheric dispersion in vertically inhomogeneous planetary boundary layers. The fractional derivative exists in a Caputo sense. We establish the maximum principle and a priori estimates for the solutions. Then, we construct a positivity-preserving finite-difference scheme, using monotone discretization in space and L1 approximation on the non-uniform mesh for the time derivative. We use appropriate grading techniques for the time–space mesh in order to overcome the boundary degeneration and weak singularity of the solution at the initial time. The computational results are demonstrated on the Gaussian fractional model as well on the boundary layers defined by height-dependent wind flow and diffusitivity, especially for the Monin–Obukhov model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a well‐balanced and positive‐preserving nonstaggered central scheme, which has second‐order accuracy on both time and spatial scales, for open‐channel flows with the variable channel width and the nonflat bottom. We perform piecewise linear reconstructions of the conserved variables and energy as well as discretize the source term using the property that the energy remains constant, so that the complex source term and the flux can be precisely balanced so as to maintain the steady state. The scheme also ensures that the cross‐sectional wet area is positive by introducing a draining time‐step technique. Numerical experiments demonstrate that the scheme is capable of accurately maintaining both the still steady‐state solutions and the moving steady‐state solutions, simultaneously. Moreover, the scheme has the ability to accurately capture small perturbations of the moving steady‐state solution and avoids generating spurious oscillations. It is also capable of showing that the scheme is positive‐preserving and robust in solving the dam‐break problem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Since the analytical solution of the stochastic age-structured human immunodeficiency virus/acquired immune deficiency syndrome model is difficult to solve, establishing an efficient numerical approximation is an important way to predict the dynamic behavior of the model. In this article, a full-discrete scheme is proposed, where the Galerkin finite element method and the positivity preserving truncated Euler–Maruyama scheme are used to discrete the age variable and the time variable, respectively. The error between the numerical solution and the analytical solution is analyzed. Finally, the theoretical results are illustrated by the numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the Black–Scholes equation is solved using a new technique. This scheme is derived by combining the Laplace transform method and the nonstandard finite difference (NSFD) strategy. The qualitative properties of the method are discussed, and it is shown that the new method is positive, stable, and consistent when low volatility is assumed. The efficiency of the new method is demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124-2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov's modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta. [ABSTRACT FROM AUTHOR]

In this paper, we propose an effcient numerical method for a comprehensive infection model that is formulated by a system of nonlinear coupling advection-diffusion-reaction equations. Using some subtle mixed explicitimplicit treatments, we construct a linearized and decoupled discrete scheme. Moreover, the proposed scheme is capable of preserving the positivity of variables, which is an essential requirement of the model under consideration. The proposed scheme uses the cell-centered finite difference method for the spatial discretization, and thus, it is easy to implement. The diffusion terms are treated implicitly to improve the robustness of the scheme. A semi-implicit upwind approach is proposed to discretize the advection terms, and a distinctive feature of the resulting scheme is to preserve the positivity of variables without any restriction on the spatial mesh size and time step size. We rigorously prove the unique existence of discrete solutions and positivity-preserving property of the proposed scheme without requirements for the mesh size and time step size. It is worthwhile to note that these properties are proved using the discrete variational principles rather than the conventional approaches of matrix analysis. Numerical results are also provided to assess the performance of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study the numerical approximation to stochastic differential equations with positive solutions. Inspired by the logarithmic truncated Euler–Maruyama method developed by Yi et al. (2021) and Lei et al. (2023), we construct a novel numerical method, called the positivity preserving logarithmic transformed truncated Milstein method, for the efficient solution of the underlying equation. Under certain assumptions, we prove the exponential integrability of the numerical solutions. On this basis, we further obtain the strong convergence and strong convergence rate of order 1 for the presented method under additional conditions. Especially, this method is applied to the stochastic tumor model with Allee effect. Several numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions. [ABSTRACT FROM AUTHOR]

In this article, one standard and four nonstandard finite difference methods are used to solve a cross-diffusion malignant invasion model. The model consists of a system of nonlinear coupled partial differential equations (PDEs) subject to specified initial and boundary conditions, and no exact solution is known for this problem. It is difficult to obtain theoretically the stability region of the classical finite difference scheme to solve the set of nonlinear coupled PDEs, this is one of the challenges of this class of method in this work. Three nonstandard methods abbreviated as NSFD1, NSFD2, and NSFD3 are considered from the study of Chapwanya et al., and these methods have been constructed by the use of a more general function replacing the denominator of the discrete derivative and nonlocal approximations of nonlocal terms. It is shown that NSFD1, which preserves positivity when used to solve classical reaction-diffusion equations, does not inherit this property when used for the cross-diffusion system of PDEs. NSFD2 and NSFD3 are obtained by appropriate modifications of NSFD1. NSFD2 is positivity-preserving when the functional relationship [ψ(h)]2=2ϕ(k){\left[\psi \left(h)]}^{2}=2\phi \left(k) holds, while NSFD3 is unconditionally dynamically consistent with respect to positivity. First, we show that NSFD2 and NSFD3 are not consistent methods. Second, we tried to modify NSFD2 in order to make it consistent but we were not successful. Third, we extend NSFD3 so that it becomes consistent and still preserves positivity. We denote the extended version of NSFD3 as NSFD5. Finally, we compute the numerical rate of convergence in time for NSFD5 and show that it is close to the theoretical value. NSFD5 is consistent under certain conditions on the step sizes and is unconditionally positivity-preserving.

In this paper, we propose and analyze a finite difference numerical scheme for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the singular chemical potential, the logarithmic term and the surface diffusion term are implicitly updated, while an explicit computation is applied to the concave expansive term. Moreover, the convective term in the phase field evolutionary equation is approximated in a semi-implicit manner. Similarly, the fluid momentum equation is computed by a semi-implicit algorithm: implicit treatment for the kinematic diffusion term, explicit update for the pressure gradient, combined with semi-implicit approximations to the fluid convection and the phase field coupled term, respectively. Such a semi-implicit method gives an intermediate velocity field. Subsequently, a Helmholtz projection into the divergence-free vector field yields the velocity vector and the pressure variable at the next time step. This approach decouples the Stokes solver, which in turn drastically improves the numerical efficiency. The positivity-preserving property and the unique solvability of the proposed numerical scheme is theoretically justified, i.e., the phase variable is always between -1 and 1, following the singular nature of the logarithmic term as the phase variable approaches the singular limit values. In addition, an iteration construction technique is applied in the positivity-preserving and unique solvability analysis, motivated by the non-symmetric nature of the fluid convection term. The energy stability of the proposed numerical scheme could be derived by a careful estimate. A few numerical results are presented to validate the robustness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

We propose a positivity-preserving finite volume scheme on non-conforming quadrilateral distorted meshes with hanging nodes for subdiffusion equations, where the differential equations have a sum of time-fractional derivatives of different orders, and the typical solutions of the problem have a weak singularity at the initial time t = 0 for given smooth data. In this paper, a positivity-preserving nonlinear method with centered unknowns is obtained by the two-point flux technique, where a new method to handling vertex unknown including hanging nodes is the highlight of our paper. For each time derivative, we apply the L1 scheme on a temporal graded mesh. Especially, the existence of a solution is strictly proved for the nonlinear system by applying the Brouwer's fixed point theorem. Numerical results show that the proposed positivity-preserving method is effective for strongly anisotropic and heterogeneous full tensor subdiffusion coefficient problems. [ABSTRACT FROM AUTHOR]

In tensor computations, tensor–vector multiplication is one of the main computational costs. We recently studied algorithms with wider applicability and more computational potential for computing the largest eigenpair of a weakly irreducible nonnegative mth-order tensor A , called higher-order Noda iteration (HONI). This method is an eigenvalue solver which uses an inner-outer scheme. The outer iteration is the update of the approximate eigenpair(s), while in the inner iteration a multilinear system has to be solved, often iteratively. For the inner iteration, we also provide a Newton-type method to solve multilinear systems, and prove that the algorithm converges to the unique solution of multilinear systems and the convergence rate is quadratic. HONI has superior performance in terms of fast convergence and positivity preserving property, and its main advantage is to use simple recursive relations to compute the approximate eigenvalue, which means that no additional tensor–vector multiplication is required. Moreover, we devise a practical relaxation criterion based on our theoretical results to improve the efficiency and practicality of HONI, called inexact HONI, and further explain the relationship between HONI and Newton–Noda iteration. Numerical experiments are provided to support the theoretical results. [ABSTRACT FROM AUTHOR]

The positivity preservation is very important in most applications solving elliptic problems. Many schemes preserving positivity has been proposed but are at most second-order convergent. Besides, in general, high-order schemes do not preserve positivity. In the present paper, we propose an arbitrary-order positivity preserving method for elliptic problems in 2D. We show how to adapt our method to the case of a discontinuous and/or tensor-valued diffusion coefficient, while keeping the expected order of convergence. We assess the new scheme on several test problems. • First arbitrary-order positivity preserving finite volume scheme on deformed meshes. • Remains high-order accurate for discontinuous and/or tensor-valued diffusion coefficient. • Assessed on several numerical tests showing the order of convergence and the positivity. [ABSTRACT FROM AUTHOR]

In this paper, we construct new positive preserving, unconditionally stable and fully discrete finite element schemes for incompressible flows with variable density. The proposed schemes employ the positive function transform ρ = F (ϕ) for the density equation and scalar auxiliary variable (SAV) q = exp (− t / T) for the momentum equation in its reformulation system. The SAV schemes are unconditionally energy stable and second-order accurate in time and lead to two decoupled generalized Stokes equations for the momentum equation to be solved at each time step. Thus, it is easy to implement and extremely efficient for these schemes when combined with an adaptive time stepping method. Numerical experiments demonstrate the stability of computations and verify the second-order accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

We consider a system of coupled equations modeling a shallow water flow with solute transport and introduce an artificial dissipation in order to improve the dissipation properties of the original cell-vertex central-upwind numerical scheme applied to these equations. Namely, a formulation is proposed which involves an artificial dissipation parameter and guarantees a consistency property between the continuity equation and the transport equation at the discrete level and, in addition, ensures the nonlinear stability and positivity of the scheme. A well-balanced positivity-preserving reconstruction is stated in terms of the conservative variable describing the concentration. We establish that constant-concentration states are preserved in space and time for any hydrodynamic flow field in the absence of source terms in the transport equation. Furthermore, we prove the maximum and minimum principles for the concentration. A suitable discretization of the diffusion term is used in combination with the proposed reconstruction procedure and artificial dissipation formulation and this allows us to prove the positivity of the concentration in the presence of diffusion effects. Finally, our numerical experiments confirm the well-balanced and positivity-preserving properties when the artificial dissipation is introduced in the central-upwind scheme, and the accuracy of the scheme for modeling surface water flows with transport processes. [ABSTRACT FROM AUTHOR]

In this work we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multiderivative Runge-Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order p > 2 is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to p = 4 and IMEX Runge-Kutta schemes of order up to p = 3. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems. [ABSTRACT FROM AUTHOR]

In this paper, the Black–Scholes equation is solved using a new technique. This scheme is derived by combining the Laplace transform method and the nonstandard finite difference (NSFD) strategy. The qualitative properties of the method are discussed, and it is shown that the new method is positive, stable, and consistent when low volatility is assumed. The efficiency of the new method is demonstrated by a numerical example.

In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H−1 gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n and p, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the \ell ^\infty bound for n and p), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

We design and analyse a non-standard finite difference (NSFD) scheme for a class of predator–prey models with non-monotonic functional response. Such functional response can be used, for instance, to model the prey's group defence ability. The numerical scheme is unconditionally dynamically consistent with respect to the positivity and boundedness of the solutions. We establish analytically the convergence of the scheme and provide numerical simulations for a Holling type-IV functional response that support the theoretical findings. The simulations further show that the NSFD scheme is more efficient than some standard numerical methods. In particular, the standard methods show dynamic inconsistencies when large step-sizes were used. On the other hand, the NSFD scheme replicated different behaviours, including the oscillatory dynamics common to predator–prey systems. [ABSTRACT FROM AUTHOR]

It is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function. [ABSTRACT FROM AUTHOR]

We know that the exact solutions for most of stochastic age-structured capital systems are difficult to find. The numerical approximation method becomes an important tool to study properties for stochastic age-structured capital models. In the paper, we study the numerical solution for stochastic age-dependent capital system based on explicit-implicit algorithm and discuss the convergence of numerical solution. For the practical significance of capital, we need to consider the positivity of the numerical solution. Therefore, we introduce a penalty factor in the stochastic age-dependent capital system to maintain the positivity, and analyze the convergence of the positive numerical solution. Finally, an example is given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn–Hilliard-type equation with Flory–Huggins–deGennes energy potential. The numerical approach is based on a convex–concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme. [ABSTRACT FROM AUTHOR]

The reactive Euler equations are a basic model for fluid flows with chemical reactions. In this work, we construct a high order positivity preserving and oscillation-free entropy stable discontinuous Galerkin (DG) scheme for the reactive Euler equations. The main ingredients of the scheme include (i) entropy preserving and entropy stable fluxes to achieve entropy stability, (ii) artificial damping terms to restrain spurious oscillations near the shocks, and (iii) positivity preserving limiters to guarantee the positivity of solutions. These ingredients are compatible with each other so that our scheme simultaneously enjoys the properties of entropy stable, oscillation-free and positivity preserving. Another distinctive feature of our scheme is that it is entropy stable for both the thermodynamic and mathematical entropies. Numerical experiments validate the designed high convergence orders of the scheme and demonstrate its good performances for discontinuous problems. • A positivity preserving and oscillation-free entropy stable DG scheme is proposed for the reactive Euler equations. • The scheme simultaneously enjoys the properties of entropy stable, oscillation-free and positivity preserving. • A distinctive feature of our scheme is that it is entropy stable for both the thermodynamic and mathematical entropies. • Numerical experiments validate the designed high convergence orders and good performances for discontinuous problems. [ABSTRACT FROM AUTHOR]