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In this work, we investigate a linear second-order numerical method for the Allen-Cahn equation with general mobility. The proposed scheme is a combination of the two-step first- and second-order backward differentiation formulas for time approximation and the central finite difference for spatial discretization, two additional stabilizing terms are also included. The discrete maximum bound principle of the numerical scheme is rigorously proved under mild constraints on the adjacent time-step ratio and the two stabilization parameters. Furthermore, the error estimates in H 1 -norm for the case of constant mobility and L ∞ -norm for the general mobility case, as well as the energy stability for both cases are obtained. Finally, we present extensive numerical experiments to validate the theoretical results, and develop an adaptive time-stepping strategy to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, based on the Lagrange multiplier method, we construct a maximum principle preserving scheme for the time-fractional Allen-Cahn equation of 2- α (0 < α < 1) order. The correction energy of this scheme is increased by a term compared to the original energy, which is O (τ α). We prove that our scheme is unconditionally stable related to the corrected energy and verify the convergence, maximum principle, and energy conservation properties of the algorithm through numerical examples. We also find that the larger the α , the faster the evolution of the time-fractional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

In this paper, a finite volume (TPFA) method is employed to simulate a degenerate breast cancer model that captures the progressive mutations from a normal breast stem cell to a tumor cell. The model incorporates a degenerate parabolic equation to represent the interaction between solid tumor growth and its environment, which involves the release of degradative enzymes governed by a partial differential equation. The discrete maximum principle is verified to be satisfied by the proposed finite volume scheme, and the convergence of the existing discrete solutions to a weak solution is proven. Additionally, the development of breast cancer is demonstrated through a numerical experiment. [ABSTRACT FROM AUTHOR]

A nonlinear vertex-centered finite volume scheme that preserves the discrete maximum principle (DMP) is proposed for anisotropic diffusion equations on polygonal meshes. The new scheme is constructed from a vertex-centered linearity-preserving scheme combined with a nonlinear correction technique. The key ingredient is the introduction of the correction coefficient and the limiter. Particularly, the choice of the limiter is discussed and two new and effective limiters are proposed. Different from most existing nonlinear DMP-preserving schemes, the new nonlinear scheme does not rely on the nonlinear combination of linear flux approximations and requires no interpolation of the auxiliary unknowns. More important is that, it can be proved theoretically that the new scheme is symmetric and coercive on general polygonal meshes with star-shaped cells. Moreover, other theoretical results such as the DMP property and stability of the correction scheme are also presented. Numerical experiments demonstrate the accuracy, efficiency, and DMP property of the new scheme on general polygonal meshes with heterogeneous and anisotropic diffusion tensors. [ABSTRACT FROM AUTHOR]

In this article, we present a hybrid numerical scheme for solving the Allen–Cahn (AC) equation on a nonuniform mesh. The AC equation represents a model for antiphase domain coarsening in a binary mixture. To solve the AC equation on nonuniform grids, the AC equation is split into linear and nonlinear terms applying the operator splitting method. As the first step, the nonlinear term is solved using the separation of variables. Next, the diffusion term is decomposed into linear operators in each space direction. In each direction, we sequentially solve each diffusion equation by applying the implicit Euler method on a nonuniform grid to update the numerical solution. Because the implicit Euler method and analytic solution do not depend on the time step size, the proposed hybrid numerical method for the AC equation on a nonuniform mesh is unconditionally stable. In addition, we prove the proposed scheme satisfies the maximum principle. To verify the superior performance of the proposed method, we conduct numerical simulations such as motion by mean curvature, total energy non-increasing property, the maximum principle and unconditional stability. [ABSTRACT FROM AUTHOR]

In this paper, by using critical point theory, we study the infinitely many solutions of partial discrete Kirchhoff type problems. Moreover, we acquire some sufficient conditions for the existence of positive solutions to the boundary value problems by using the strong maximum principle. As far as we know, this is the research of partial discrete Kirchhoff type problems for the first time. Finally, our major results are explained with four examples. [ABSTRACT FROM AUTHOR]

Two new numerical schemes to approximate the Cahn-Hilliard equation with degenerate mobility (between stable values 0 and 1) are presented, by using two different non-centered approximation of the mobility. We prove that both schemes are energy stable and preserve the maximum principle approximately, i.e. the amount of the solution being outside of the interval [ 0 , 1 ] goes to zero in terms of a truncation parameter. Additionally, we present several numerical results in order to show the accuracy and the well behavior of the proposed schemes, comparing both schemes and the corresponding centered scheme. [ABSTRACT FROM AUTHOR]

We present a new energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn (NtAC) model, where the standard Laplace operator is deliberately replaced with a spatial convolution term that aims at describing long-range interactions among particles and the structure-preserving scheme is obtained based on the operator splitting method. The discrete maximum bound principle and global convergence of the new scheme are analyzed rigorously. Moreover, the scheme is strictly dissipative for a modified energy, which coincides with the original energy up to O (τ) , where τ is the time step. To the best of our knowledge, this is the first work to show that the operator splitting method can guarantee the maximum bound principle for a ternary Allen-Cahn model. Various 2D/3D numerical experiments including the NtAC model with mass conservation are tested to verify the bound-preserving and energy-stable properties of the resulted scheme. [ABSTRACT FROM AUTHOR]

We utilize the second-order quadrature-based finite difference method and the high-order parametric integrating factor Runge-Kutta (pIFRK) integrators to construct efficient and accurate schemes for solving the nonlocal Allen-Cahn equation. These schemes preserve the maximum principle for any time-step, and exhibit up to fourth-order accuracy in the temporal direction. We establish a rigorous error estimate and an asymptotic compatibility analysis for the pIFRK schemes. Numerical experiments demonstrate the accuracy and structure-preserving property of the proposed schemes, verify their asymptotic compatibility, and investigate the discontinuity of the nonlocal Allen-Cahn equation under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method to solve the space fractional Allen-Cahn equation. First, to remove the constraint of time step, an efficient scheme is constructed by the stabilization method, where a first order method is used in time direction and a second-order weighted and shifted Grünwald difference formula is applied in spatial direction. Then, the unique solvability of the scheme is given for any time step. Meanwhile, the unconditional discrete maximum principle preserving and energy stability are rigorously proved. And we also give a detailed error analysis. Besides, to reduce computational time and improve computational efficiency, adaptive time step strategy is used when simulating two important properties and coarsening dynamics. Finally, some numerical experiments show that adaptive time step strategy is effective in long time simulations. [ABSTRACT FROM AUTHOR]

In this paper, based on the mass-lumping finite element space discretization, we incorporate the integrating factor Runge–Kutta method and stabilization technique to develop a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen–Cahn equation and its conservative forms. The proposed methods are linear, without requiring any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Several numerical experiments verify the high-order temporal accuracy of the proposed schemes, as well demonstrate the ability to preserve the maximum principle, mass conservation, and energy stability over long periods. Moreover, by the aid of numerical simulation, we show that the proposed schemes also have good performances in terms of structure-preserving with high order finite element method. [ABSTRACT FROM AUTHOR]

We establish Itô's formula along flows of probability measures associated with general semimartingales; this generalizes existing results for flows of measures on Itô processes. Our approach is to first establish Itô's formula for cylindrical functions and then extend it to the general case via function approximation and localization techniques. This general form of Itô's formula enables the derivation of dynamic programming equations and verification theorems for McKean–Vlasov controls with jump diffusions and for McKean–Vlasov mixed regular–singular control problems. It also allows for generalizing the classical relationship between the maximum principle and the dynamic programming principle to the McKean–Vlasov singular control setting, where the adjoint process is expressed in terms of the derivative of the value function with respect to the probability measures. [ABSTRACT FROM AUTHOR]

We develop a class of explicit unconditionally maximum-principle-preserving multi-step methods for the Allen-Cahn-type semilinear parabolic equations. Based on the usual second-order finite difference space discretization, we first show that the strong-stability-preserving (SSP) integrating factor multi-step method can preserve the maximum-principle under the condition τ ≤ C τ F E , where C is the SSP coefficient and τ F E is independent of the space step size, thus avoiding the strict parabolic Courant-Friedrichs-Lewy condition τ = O (h 2). By introducing the stabilization technique and replacing the exponential functions with two different approximations, we obtain two types of improved stabilized integrating factor multi-step (isIFMS) schemes. The unconditional maximum-principle-preservation, mass-conservation, linear stability, and error estimates are analyzed for the isIFMS schemes. Since explicit SSP multi-step methods have no order barrier, the proposed framework offers a concise but effective approach for developing temporal high-order, unconditionally maximum-principle-preserving schemes for Allen-Cahn-type equations, and mass-conserving schemes for their conservative reformulations. Finally, numerical experiments validate theoretical results of the proposed isIFMS schemes. [ABSTRACT FROM AUTHOR]

In this work, we develop the numerical analysis of a parabolic system of PDE describing the proliferation-invasion structure of hypoxic glial cells in the brain. Explicitly, we propose a fully discrete numerical scheme, based on a semi-implicit Euler discretization in time and Finite Element (FE) discretization on space, for approximating the solutions of the continuous model. For this numerical scheme we prove some properties including well-posedness, positivity, maximum principle, uniform estimates, convergence towards weak solutions, optimal error estimates and asymptotic behaviour of the discrete solutions. Finally, we present some numerical simulations which validate the theoretical results. [ABSTRACT FROM AUTHOR]

A fourth-order finite difference scheme for the Kuramoto-Tsuzuki equation is considered. The theoretical properties of the proposed scheme, such as the existence, uniqueness and the consistency errors are analyzed. The error estimates in a discrete maximum-norm show that the convergence rates of the difference scheme are of order O (h 4 + k 2). Some numerical experiments are reported to confirm the advantages of the proposed difference scheme by comparing it with other existing recent numerical methods. [ABSTRACT FROM AUTHOR]

This paper presents a systematic methodology, called the MBP-preserving iteration technique, to develop the maximum bound principle (MBP) preserving numerical algorithms for a class of semilinear parabolic equations. Two types of MBP-preserving iterations are suggested to solve two well-known θ -weighted schemes, respectively. Within some mild time-step constraints, the corresponding iteration solutions are proved to preserve the MBP property at each iteration step so that the numerical scheme has a uniquely MBP-preserving solution. In addition, concise error estimates in the maximum norm are established on nonuniform time meshes. Several numerical examples coupled with an adaptive time-stepping strategy are implemented for the Allen-Cahn model to confirm the theoretical findings and demonstrate their effectiveness for long-time numerical simulations. [ABSTRACT FROM AUTHOR]

We prove two versions of the classical Aleksandrov-Bakelman-Pucci (ABP) maximum principle using norms over contact sets for L p -viscosity sub/super-solutions of fully nonlinear uniformly elliptic equations F (x , u , D u , D 2 u) = f (x) in Ω ⊂ R n , with measurable and unbounded terms. More precisely, we assume that the structural coefficient function γ (corresponding to the drift coefficient for linear equations) is in L n (Ω). Such a result was previously known for L n -viscosity solutions only when γ was bounded. We use the ABP maximum principle to prove pointwise properties of L p -viscosity solutions for equations with unbounded γ and extend the theory of L p -viscosity solutions to the case of equations with γ ∈ L n (Ω). We use a recent generalization of ABP maximum principle by Krylov [28]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the linear quadratic decentralized control of forward and backward stochastic difference system (FBSDS) with multiplicative noises and nested asymmetric information. The main contribution is to give the equivalent condition for the solvability of the problem and the explicit decentralized optimal controllers in terms of Riccati equations. The key technology is solving the forward and backward stochastic difference equations with partial information derived from stochastic maximum principle. • LQ decentralized control of forward and backward stochastic difference system with nested asymmetric information. • Forward and backward stochastic difference equations with partial information. • Present solvability condition and explicitly decentralized optimal controllers of the problem. • Propose novel Riccati equations to solve forward and backward stochastic difference equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an efficient maximum bound principle (MBP) preserving and mass conservative projection method for the conservative Allen–Cahn equation. The proposed projection operator can be proven contractive in the discrete L 2 norm. Discrete MBP and mass conservation are rigorously presented for a second-order exponential time differencing scheme with a central difference discretization in space. Various numerical examples are performed to verify these theoretical results and demonstrate the accuracy and robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a novel numerical algorithm for efficient modeling of three-dimensional shape transformation governed by the modified Allen-Cahn (A-C) equation is developed, which has important significance for computer science and graphics technology. The new idea of the proposed method is as follows. Firstly, the operator splitting method is used to decompose the three-dimensional problem into a series of one-dimensional subproblems that can be solved in parallel in the same direction. Secondly, a temporal p-adaptive strategy, which is based on the extrapolation technique, is proposed to improve the convergence order in time and preserve the computational efficiency simultaneously. Finally, a parallel least distance modification technique is developed to force the discrete maximum bound principle. The proposed method achieves high precision and high efficiency at the same time. Numerical examples include the effectiveness of the p-adaptive method and the bound preserving least distance modification, and a series of complex three-dimensional shape transformation modelings. • The operator splitting method is used to solve 3D shape transformation PDE. • A temporal p-adaptive strategy is developed to improve computational efficiency. • A least-distance modification is developed to force the discrete maximum bound. [ABSTRACT FROM AUTHOR]

The article presents a new approach for optimizing the trajectories of spacecraft with finite thrust. The recently proposed angular variable – the auxiliary longitude – is considered as an independent variable. The advantages of using this variable to optimize a trajectory with a fixed angular distance and free transfer duration are shown. A complete system of necessary optimality conditions for the considered representation of the optimal control problem is presented. An approach to solving the minimum-fuel problem is considered that is based on the sequential solution of the problems of optimizing the limited power trajectories and the trajectories with a constant exhaust velocity. This approach includes verifying the existence of a solution to the minimum-fuel problem by solving the thrust minimization problem. We present a method for optimization of the finite-thrust trajectories with constant exhaust velocity, based on the application of the maximum principle and the continuation method, which does not require the user to specify an initial guess for the initial values of costate variables. The article presents numerical examples to illustrate the application of such approach, and discusses the properties of the obtained optimal trajectories. • Using of auxiliary longitude as an independent variable. • Solving the trajectory optimization problem using maximum principle and continuation technique. • Automated trajectory optimization of spacecraft with limited-power or constant exhaust velocity thruster. • Method to verify the existence of finite-thrust trajectory. • Existence of minimum of optimal transfer duration with respect to the thrust value. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new nonlinear conservative and maximum-principle-preserving finite volume element method for heterogeneous anisotropic diffusion equations on triangular or strictly convex quadrilateral meshes. First, we decompose the numerical flux of the standard finite volume element method on each dual edge into two parts: main part having a two-point-flux structure, and residual part owning a multi-point-flux structure. Then, according to the sign of the residual part and the distribution of the local extremum, the residual part is modified to a two-point-flux structure with a nonlinear term. The resulting nonlinear scheme not only inherits conservation and accuracy of original scheme, but also possesses a maximum-preserving structure without extra restrictions on the meshes and diffusion coefficients. Several numerical experiments verify that our nonlinear scheme has an optimal convergence order and satisfies the discrete maximum principle. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical method for generalized Allen–Cahn equation with nonlinear mobility and convection term. We propose a linear second-order finite difference scheme which preserves the discrete maximum bound principle (MBP). The scheme is discretized by stabilized Crank–Nicolson in time, upwind scheme for convection term and central-difference scheme for diffusion term. We show that the proposed scheme preserves the discrete MBP under some constraints on temporal step size and stabilizing parameter. Optimal L ∞ -error estimate is obtained for our scheme. Several numerical experiments are performed to validate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper explores the use of the numerical inverse hanging method to optimize spatial truss structures. It proposes an optimization algorithm and establishes a corresponding numerical model. The algorithm consists of a form-finding algorithm based on the numerical inverse hanging method and a topology optimization algorithm that uses the maximum stress design principle. Component length optimization considers the characteristics of cross-sections and internal forces. The cross-sectional area of the component is automatically determined based on internal forces and design strength, enhancing the stable load-bearing capacity of the components. The model is optimized using the algorithm, resulting in a significant increase in axial forces in the internally effective truss members. Bending moments have a negligible impact compared to axial forces, minimally affecting the structure. The study analyzes form-finding results for various ground structures within the same design domain, considering the impact of different boundary conditions on the final outcomes. The results indicate that altering the ground structure does not affect the topology of the spatial truss structure. However, changing the boundary conditions leads to variations in the final form-finding results, affirming the stability and robust applicability of the algorithm. This algorithm offers an efficient and precise approach to form-finding challenges in spatial truss structures, presenting novel perspectives for further research in this domain. • An optimization design algorithm suitable for form-finding of spatial truss structures is proposed. • The proposed algorithm can perform topology optimization while minimizing bending moments. • Influence of complex constraint on the ultimate form-finding result is considered. [ABSTRACT FROM AUTHOR]

In this paper, we construct a second-order accurate, energy stable and maximum bound principle-preserving scheme for the Allen–Cahn equation with a general mobility based on the stabilized exponential scalar auxiliary variable (SESAV) approach. Some extra stabilizing terms are added to the discretized scheme for the purpose of improving numerical stability. We first proved the maximum bound principle (MBP) under reasonable constraints on time step size and the stabilization parameter. Then, we found that the proposed scheme is unconditionally energy-stable. Finally, a numerical example is carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Let G = (V (G) , E (G)) be a graph with vertex set V (G) and edge set E (G). The line graph L G of G is the graph with E (G) as its vertex set and two vertices of L G are adjacent in L G if and only if they have a common end-vertex in G. The resistance distance R G (x , y) between two vertices x , y of G is equal to the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor. The resistance diameter D r (G) of G is the maximum resistance distance among all pairs of vertices of G. In this paper, it was shown that the resistance diameter of the line graph of a tree or unicyclic graph is no more than that of its initial graph by utilizing series and parallel principles, the principle of elimination and star-mesh transformation in electrical network theory. And experiment also indicated that the inequality D r (L G) ≤ D r (G) is true for every simple nonempty connected graph G with less than 12 vertices. Thus it was conjectured that D r (L G) ≤ D r (G) for every simple nonempty connected graph G. [ABSTRACT FROM AUTHOR]

In this article, we investigate some of the fine properties of the value function associated with an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems. [ABSTRACT FROM AUTHOR]

• We present numerical simulations of dynamic elastoplastic plates by using the non-incremental (BEN) principle. • Unlike the standard incremental procedure, the BEN principle characterizes the entire trajectory. • Two kinds of plates: the Love-Kirchhoff and the Reissner-Mindlin circular plates undergoing uniform pressure are studied. • The analytical results are compared against numerical one derived by the FEM. This paper is devoted to the numerical simulations of elastoplastic plates at small strains by using the non-incremental Brezis-Ekeland-Nayroles (BEN) principle. This space-time variational principle is based on the energy dissipation and transforms the boundary value problem into a null minimization one under constraints. Unlike the standard incremental procedure, the BEN principle characterizes the entire trajectory and delivers the mechanical fields at all time steps simultaneously. Two kinds of plates: the thin plate Love-Kirchhoff and the thick Reissner-Mindlin circular plates undergoing uniform pressure are studied respectively. The description of the dicretization of the principle using the mixed finite element method and the implementation are detailed. The obtained results are compared against analytical solutions provided in the literature and numerical predictions derived by the classical incremental procedure. [ABSTRACT FROM AUTHOR]

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem − ∇ 2 u = f in Ω ∖ Γ with Dirichlet boundary condition such that f is smooth in Ω ∖ Γ and the jump functions [ u ] and [ ∇ u ⋅ n → ] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of − ∇ 2 u = f + g δ Γ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇ u ⋅ n → are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω + and Ω −. The coefficient matrix A in the resulting linear system A x = b , following from the proposed scheme, is independent of any source term f , jump condition g δ Γ , interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω + or Ω −. The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources. [ABSTRACT FROM AUTHOR]

When using a Bayesian network to model a practical problem, weak prior knowledge projected as ancestral constraints is necessary. However, it is difficult to directly utilize these non-decomposable constraints using search strategies based on the decomposable score. In this study, we attempt to solve this problem by conducting an implicate path-space search graph and driving the A* algorithm, which is used to obtain the globally optimal solution satisfying the given constraints. We use a maximum covering principle to provide useful pruning rules based on these constraints in the new framework. Moreover, we improve the simple heuristic and the static k-cycle conflict heuristic to adapt to ancestral constraints. We theoretically prove that the new heuristic functions remain admissible and consistent. Our experiments demonstrate that the proposed framework with the new heuristic functions significantly reduces the space complexity of A* search compared with state-of-the-art frameworks, such as Bayesian network graphs and equivalent class trees, when integrating ancestral constraints. [ABSTRACT FROM AUTHOR]

[Display omitted] • A novel mathematical model of Corona virus with new hybrid fractional operator derivative are presented. • Three control variables are presented to minimize the number of infected population. • Necessary control conditions are derived. • Two numerical methods are constructed to study the behavior of the obtained fractional optimality system. • The stability of the proposed methods are proved. • Numerical simulations and comparative studies are given. Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann–Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo's fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grünwald-Letnikov nonstandard finite difference method. The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin's maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grünwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists. [ABSTRACT FROM AUTHOR]

This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated. [ABSTRACT FROM AUTHOR]

• A novel lattice structure is constructed by translating, rotating, and splicing. • Bending defects of translational triangles can regulate the mechanical behavior. • Effects of bending angles, aspect ratio, and relative density are studied. • A theoretical model of plateau stress at the given aspect ratio is established. Lattice structures have been proven to be one of the most promising metamaterials since they can achieve different mechanical properties by tuning the topology of the microstructure. A novel lattice structure is constructed by Rotating triangles Translated out of plane and Splicing each other (RTS) inspired by the topology strategies of rotating and intertwining. The bending defect in the translational triangles is employed to adjust the mechanical behavior to meet the complicated service environment. The structures under different bending angles were fabricated by selective laser melting technology using orientation-insensitive Ti6Al4V material. The quasi-static compression experiments and numerical simulations show that the bending angle can significantly regulate the elastic modulus, yield strength, and plateau stress of the structure. The initial peak stress (i.e. equivalent yield strength) of RTSs can be adjusted to a level comparable to plateau stress through the bending defect, which is meaningful for anti-collision protection. The elastic modulus, the yield strength, and the plateau stress are inversely proportional to the equivalent aspect ratio (a / h) besides the plateau stress of a / h < 0.7. The effect of relative density on the mechanical properties of RTSs is discussed through the Gibson-Ashby model. This suggests the relative elastic modulus is dominated by the stretching behavior of struts, while the relative yield strength is dominated by the bending behavior of struts. Furthermore, a theoretical model of plateau stress is established based on the principle of plastic hinges, and the maximum error with the numerical simulation is less than 11 % at 1 % ≤ ρ ¯ ≤ 10 % and a / h = 1. Altogether, this novel lattice structure has favorable elastic–plastic properties compared with the common lattice structures. [ABSTRACT FROM AUTHOR]

In this paper, a class of non-negativity-preserving and maximum-principle-satisfying finite difference methods have been derived by Vieta theorem for one-dimensional and two-dimensional Fisher's equation. By using the positivity and boundedness of numerical and exact solutions, it is shown that numerical solutions obtained by current methods converge to exact solutions with orders of O (Δ t + (Δ t / h x) 2 + h x 2) for one-dimensional case and O (Δ t + (Δ t / h x) 2 + (Δ t / h y) 2 + h x 2 + h y 2) for two-dimensional case in the maximum norm, respectively. Here, Δ t , h x and h y are meshsizes in t -, x - and y -directions, respectively. Finally, numerical results verify that the proposed method can inherit the monotonicity, boundedness and non-negativity of the continuous problems. • New Du Fort-Frankel methods are devised for Fisher's equation. • They are non-negativity-preserving and maximum-principle-satisfying schemes. • Errors in maximum norm are given for them. • They are very good at long-term simulations. • They are very easy to be implemented. [ABSTRACT FROM AUTHOR]

In this paper, we develop a predictor–corrector Monte Carlo method for solving the thermal radiative transfer equations. A two-step numerical method is employed when the material temperature violates the maximum principle. It utilizes the effective scattering technique in the standard implicit Monte Carlo (IMC) method in each step. Then a linear combination of the prediction and correction solutions is obtained as the final solution, where the coefficient is determined by solving an optimization problem. The new method is more implicit than the IMC method in that it uses a more implicit estimate for the material temperature dependent quantities. Moreover, we note that the last modification step is necessary because using the iterative solution directly may overestimate the material temperature when large time steps are used. Numerical simulations show that the new method can significantly mitigate the violation of maximum principle in comparison with the standard IMC method, and results in more accurate solutions than the original predictor–corrector method when the time step is large. • A predictor-corrector Monte Carlo method is proposed for the TRT equations. • The final solution is a linear combination of the prediction and correction solutions. • The new method can mitigate the violation of maximum principle in IMC. • The new method results in more accurate result than the original predictor–corrector method. [ABSTRACT FROM AUTHOR]

The paper addresses the problem of transient performance improvement of direct model reference adaptive control (MRAC) of discrete linear time-invariant (LTI) plants. Two solutions to the problem are proposed and use the certainty equivalence principle and the idea of regressor recording over a past period of time. The first solution is based on the principle of augmented error , while the second one uses new scheme of high order tuner generating predicted values of adjustable parameters. The recording of the past regressor is provided by application of a special SISO filter (linear operators with "memory") to the closed-loop error model. It is proven and demonstrated by numerical examples that the proposed solutions can provide asymptotic (not exponential) convergence of the adjustable parameters under some simple condition which is weaker than the persistent excitation one. The proposed solution can be considered as a generalization of the identification/adaptation algorithm proposed by Kreisselmeier for continuous systems [21, 22]. [ABSTRACT FROM AUTHOR]

• First high order MPP scheme for Allen-Cahn type equations. • Two widely applied techniques are adopted to ameliorate the restriction on time step size caused by the MPP property. • The proposed method is explicit, easy to complement and also has good accuracy. Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn equation has been an open problem in recent years. This work provides a positive answer by designing and analyzing a class of up to fourth order maximum principle preserving integrators for the Allen-Cahn equation. First, the second order finite difference discretization is applied to the Allen-Cahn equation in the space direction. The obtained semi-discrete system also preserves the maximum principle and the energy dissipation law. Then the fully discrete numerical scheme is obtained by applying the Lawson transformation and the Runge-Kutta integration in the time direction. We define sufficient conditions for explicit integration factor Runge-Kutta scheme to preserve the maximum principle, namely, the Shu-Osher form of the underlying Runge-Kutta scheme has non-negative coefficients α i , j , nondecreasing abscissas c i and the time step size τ > 0 satisfies τ { β i , j α i , j } ∈ − 4 , 1 2 . We prove that the proposed method is convergent with order O (τ p + h 2) in the discrete L ∞ norm. A fast solver is then applied to the discrete system to accelerate numerical computations. Various experiments for 1D, 2D and 3D problems are provided to illustrate the high-order convergence and maximum principle preserving of the proposed algorithms over a long time and verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a finite volume scheme preserving discrete maximum principle (DMP) for the imperfect interface problem. We prove the existence of the numerical solution and prove that when the interface parameter tends to 0, the numerical solutions of imperfect interface problem have a subsequence converging to the numerical solution of perfect interface problem. We apply a domain decomposition method called Robin-Robin iteration to solve the scheme and prove the DMP on each iteration step. The numerical experiments show the DMP-preserving property and accuracy of this scheme. [ABSTRACT FROM AUTHOR]

• Derived the explicit Fokker-Planck equations for the SDE with multiplicative α stable noise. • Constructed a new numerical scheme for solving nonlocal Fokker-Planck equation. • Applied the results to a nonlinear filtering problem. • Results are expected to assist simulation study of, for example, the stochastic climate dynamics. In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α -stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem. [ABSTRACT FROM AUTHOR]

A result by Courrège says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form L = L σ , b + L μ where L σ , b u (x) = tr (σ σ T D 2 u (x)) + b ⋅ D u (x) and L μ u (x) = ∫ R d ∖ { 0 } (u (x + z) − u (x) − z ⋅ D u (x) 1 | z | ≤ 1) d μ (z). This class of operators coincides with the infinitesimal generators of Lévy processes in probability theory. In this paper we give a complete characterization of the operators of this form that satisfy the Liouville theorem: Bounded solutions u of L u = 0 in R d are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of L u = 0 in R d. The proofs combine arguments from PDEs and group theory. They are simple and short. [ABSTRACT FROM AUTHOR]

Comparison principles are developed for piecewise linear finite element approximations of quasilinear elliptic partial differential equations. We consider the analysis of a class of nonmonotone Leray-Lions problems featuring both nonlinear solution and gradient dependence in the principal coefficient, and a solution dependent lower-order term. Sufficient local and global conditions on the discretization are found for conforming finite element solutions to satisfy a comparison principle, which implies uniqueness of the solution. For problems without a lower-order term, our analysis shows the meshsize is only required to be locally controlled, based on the variance of the computed solution over each element. We include a discussion of the simpler semilinear case where a linear algebra argument allows a sharper mesh condition for the lower order term. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical study for the multi-dimensional fractional-in-space Allen-Cahn equation with homogeneous Dirichlet boundary condition. By utilizing Strang's second-order splitting method, at each time step, the numerical scheme can be split into three sub-steps. The first and third sub-steps give the same ordinary differential equation, where the solutions can be obtained explicitly. While a multi-dimensional linear fractional diffusion equation needs to be solved in the second sub-step, and this is computed by the Crank-Nicolson scheme together with alternating directional implicit (ADI) method. Thus, instead of solving a multidimensional nonlinear problem directly, only a series of one-dimensional linear problems need to be solved, which greatly reduces the computational cost. A fourth-order quasi-compact difference scheme is adopted for the discretization of the space Riesz fractional derivative of α (1 < α ≤ 2). The proposed method is shown to be unconditionally stable in L 2 -norm, and satisfying the discrete maximum principle under some reasonable time step constraint. Finally, numerical experiments are given to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, a new splitting technique is implemented for solving hyperbolic PDEs. As the main result, the new methods preserve the maximum principle unconditionally or with a mild condition on discretization parameters in comparison with well known methods. Damping of numerical solution in time evolution is investigated. For numerical solution of the Burgers' equation, as a nonlinear problem, an iterative method based on the new splitting technique is presented. Efficiency of the new methods is examined via numerical examples. [ABSTRACT FROM AUTHOR]