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We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove conditional stability and derive 4th-order error bound in the enlarged energy norm. It exploits additional sought functions which approximate 2nd-order non-mixed spatial derivatives of the exact solution. At the first time level, a similar two-level in time scheme is applied, without using derivatives of the data in the problem. No iterations are required to implement the scheme. We also present results of various 3D numerical experiments. They demonstrate a very high accuracy of the scheme, for smooth data, and its advantages in the error behavior over the classical explicit 2nd-order scheme, for nonsmooth data. An example of the wave in a layered medium (with a discontinuous speed of sound) initiated by the Ricker-type wavelet source function is also included. [ABSTRACT FROM AUTHOR]

The paper demonstrates constructing monotone difference scheme for the two-dimensional (in space) transfer equation in divergent form by the Lagrangian–Eulerian approach. It is based on the use of spatial grids formed by the approximate traces of the characteristics for this operator. The use of such special grids made it possible to circumvent the limitation by S. K. Godunov on the absence of monotone schemes above the first order of convergence. A brief theoretical justification for the second order of convergence and supporting results of the computational experiment are presented. [ABSTRACT FROM AUTHOR]

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time-dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable with a mixed finite element method in spatial variables. The focus of the study is to analyze stability results and to establish optimal error estimates, up to a logarithmic factor, for both the solution and the flux in L 2 -norm when the initial data u 0 ∈ H 0 1 (Ω) ∩ H 2 (Ω) . Additionally, an error estimate in L ∞ -norm is derived for 2D problems. All the derived estimates and bounds in this article remain valid as α → 1 - , where α is the order of the Caputo fractional derivative. Finally, the results of several numerical experiments conducted at the end of this paper are confirming our theoretical findings. [ABSTRACT FROM AUTHOR]

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ2 + h2) in L∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ4 + h4) in L∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates a higher-order numerical technique for solving an inhomogeneous time fractional reaction-advection-diffusion equation with a nonlocal condition. The time-fractional operator involved here is the Caputo derivative. We discretize the Caputo derivative by an L1–2 formula, while the compact finite difference scheme approximates the spatial derivatives. The numerical approach is based on Taylor's expansion combined with modified Gauss elimination. A thorough study demonstrates that the suggested approach is unconditionally stable. Tabular results show that the proposed scheme has fourth-order accuracy in space and (3 - β) -th-order accuracy in time. The numerical results of two test problems demonstrate the effectiveness and reliability of the theoretical estimates. [ABSTRACT FROM AUTHOR]

DiPerna and Lions (Invent Math 98(3):511–547, 1989) established the existence and uniqueness results for weak solutions to linear transport equations with Sobolev velocity fields. Motivated by fluid mechanics, this paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is L p -strongly convergent in the class of DiPerna–Lions weak solutions. The second method is based on an implicit scheme with L 2 -estimates, where the discrete Helmholtz–Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and L 2 -strongly convergent in the class of DiPerna–Lions weak solutions. The key point for both of the methods is to obtain fine L 2 -bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna–Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method for sharp interfaces involving transport equations, where rigorous discrete approximation of level-sets and their geometric quantities is discussed. [ABSTRACT FROM AUTHOR]

In this paper, an Arrow-Hurwicz iterative finite element method based on charge-conservation for solving the stationary inductionless magnetohydrodynamic equations is proposed and analyzed. The current density and electric potential are discretized by H (div , Ω) × L 2 (Ω) -conforming finite element pairs, respectively, which maintains charge-conservation. The hydrodynamic unknowns are discretized by stable velocity-pressure finite element pairs. Furthermore, we use the Arrow-Hurwicz iterative method to decouple the velocity and pressure, which leads to that there is no large-scale saddle point system that needs to be solved at each iteration step except for the determination of the initial functions. It has been proven that the proposed method converges geometrically with a contraction number independent of the mesh width. Numerical results are presented to verify the results of theoretical analysis and the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper proposes a family of weighted- θ high-order ADI schemes for sine-Gordon equations in high dimensions in combination with the discretization of a three-level linearized scheme in temporal direction and a classical compact difference scheme in spatial direction. The unique solvability, convergence and stability of the difference scheme in two dimensions are strictly proved with the convergence order four in space and order two in time under the L ∞ -norm. More remarkably, the present numerical scheme can be extended to cope with the three-dimensional case feasibly. Several benchmark problems in two and three dimensions verify theoretical results and the capacity in simulating the dynamic behaviour of ring solitons. Specially, several novel numerical isosurfaces in three dimensions are presented for the first time. [ABSTRACT FROM AUTHOR]

A spatial two-grid compact difference method for the nonlinear Volterra integro-differential equations with the Abel kernel is proposed to reduce the computational cost and improve the accuracy of the scheme. The proposed scheme firstly solves a small nonlinear compact finite difference system on a coarse grid and then solves a large linear compact finite difference system on a fine grid based on the coarse-grid solution using Newton's linearization and the higher-order mapping operator. Then, we combine the properties of the higher-order mapping operator with the two-grid analysis method as well as the singularity of the solutions to prove the stability and convergence of the proposed algorithm under the L 2 -norm with the order O (τ 2 + H 8 + h 4) . Moreover, the constructed method is extended to the two-dimensional case. Several numerical experiments verify the effectiveness of the proposed method and show its competitiveness compared with the existing methods. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical approach for solving linear Fredholm integro-differential equations. We integrate the backward finite difference method with the Nyström method to reduce the system size by half compared to the method proposed by Tair et al. (J. Appl. Math. Comput. Mech. 20(3):53-64, 2021). This reduction enhances computational efficiency and is particularly advantageous for large integration intervals. Our method ensures convergence by constructing a new norm for R N + 1 . Numerical tests demonstrate the superiority of our method in terms of execution time and accuracy. The results contribute to the ongoing efforts in solving integro-differential equations more effectively, offering a robust tool for applications in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct optimal error estimates of second-order stabilized scheme for the incompressible magnetohydrodynamic(MHD) system. For this purpose, we first construct first- and second-order semi-discrete schemes in which the time derivative term is treated by the first-order backward Euler method and the second-order backward difference formulation, respectively. Moreover, the nonlinear terms are treated by semi-implicit method, and the coupling of velocity and pressure is decoupled by a Gauge–Uzawa method. Thus, the schemes achieve decoupling of velocity, magnetic field and pressure. Most importantly, the proposed schemes do not need to deal with the artificial boundary conditions on pressure and do not need to give an initial value of pressure. Then, the unconditional stability of the two schemes is demonstrated. Furthermore, through rigorous error analysis, we provide optimal convergence orders for all unknowns. Finally, some numerical experiments demonstrate the accuracy and effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

This paper deals with the application of probabilistic time integration methods to semi-explicit partial differential–algebraic equations of parabolic type and its semi-discrete counterparts, namely semi-explicit differential–algebraic equations of index 2. The proposed methods iteratively construct a probability distribution over the solution of deterministic problems, enhancing the information obtained from the numerical simulation. Within this paper, we examine the efficacy of the randomized versions of the implicit Euler method, the midpoint scheme, and exponential integrators of first and second order. By demonstrating the consistency and convergence properties of these solvers, we illustrate their utility in capturing the sensitivity of the solution to numerical errors. Our analysis establishes the theoretical validity of randomized time integration for constrained systems and offers insights into the calibration of probabilistic integrators for practical applications. [ABSTRACT FROM AUTHOR]

In this paper, the superconvergence error analysis of an energy-conserving Galerkin fully discrete scheme is proposed and investigated for the two-dimensional sine-Gordon equation. The unique solvability of the numerical scheme as well as the energy conservation are studied firstly. Then, based on the special property of the bilinear element on the rectangular mesh and the superclose estimate between interpolation and Ritz projection of the exact solution in $ H^1 $ H 1 -norm, the global superconvergence result in $ H^1 $ H 1 -norm is obtained in terms of a post-processing technique. Finally, numerical results are provided to confirm the energy conservation and superconvergence properties. [ABSTRACT FROM AUTHOR]

In this study, a novel meshless kernel-based approximation method is developed for solving the Sobolev equation. Radial kernels are employed as approximating basis functions within the least-squares framework, enabling spatial discretization on trial spaces spanned by translates of positive-definite radial kernels. The interior and boundary data are then decomposed, allowing the conversion of the underlying differential algebraic equations into a system of first-order ordinary differential equations (ODEs) for interior data only. Finally, an efficient and high-order ODE solver is invoked for stable time integration. The proposed meshless scheme is a low-cost computational scheme that circumvents the ill-conditioning and improves eigenvalue stability. The method performance is evaluated on a two-dimensional Sobolev equation over rectangular and complex-shaped domains with regular and scattered nodes. Some recommendations are made for resolving discontinuous and singular solutions. [ABSTRACT FROM AUTHOR]

The Allen–Cahn equation models antiphase domain coarsening in binary mixtures. To capture more complex and natural dynamics, we derive the anisotropic Allen–Cahn (aAC) equation, incorporating anisotropic mobility from an energy functional with a matrix-valued mobility. We present properties of the aAC equation, including energy dissipation and linear stability analysis, using this energy functional. The semi-discretized aAC equation is studied for properties like unique solvability, the maximum principle, and unconditional energy-gradient stability in a time-discrete manner. Additionally, numerical experiments demonstrate properties such as the maximum principle, unconditional energy stability, linear stability analysis, total area decreasing property, and transition layer profile. [ABSTRACT FROM AUTHOR]

This paper presents two high-order compact difference schemes to discuss the numerical solution of the one-dimensional and two-dimensional multi-term time-fractional convection-diffusion equation of the Sobolev type based on the Caputo fractional derivative. For this purpose, we employ the L2 formula for the temporal discretization of the Caputo fractional derivatives and introduce a new compact difference operator for the space discretization. The proposed schemes transform the original problem into a system of algebraic equation. We present a novel analysis of the convergence and theoretical stability of both methods. The difference schemes have fourth-order and 3 - max { α r , β r } order of accuracy in space and time respectively, where α r and β r are the orders of the fractional derivatives in the multi-term convection-diffusion model. Additionally, we construct an L2-type numerical scheme on non-uniform graded meshes to address problems where the solution exhibits weak initial regularity. Some test functions are provided to demonstrate the accuracy and efficiency of the proposed schemes. The numerical results of inclusive examples confirm the proposed schemes’ theoretical results and illustrate their applicability and efficiency. [ABSTRACT FROM AUTHOR]

The component-wise WENO methods are very favorable for simulating complex problems without full characteristic structures in closed form, as they avoid the costly and dependency characteristic-decomposition process which is strictly required by their characteristic-wise counterparts. However, unfavorable numerical oscillations can be induced by the component-wise WENO methods, especially for those with very low dissipations. This study tries to overcome this crucial shortcoming existing in the component-wise WENO-ACM method, whose characteristic-wise counterpart was well-validated and proved to be highly efficient. The present analysis indicates that the numerical oscillations are basically due to the fact that the mapping function of WENO-ACM over-amplifies the contributions of less-smooth substencils. Therefore, a practical principle for designing the mapping function is proposed to prevent the numerical oscillations: the relative order of the nonlinear weights obtained from the classical WENO-JS method must be guaranteed. Obeying this principle, we present an effective but easy-implemented mapping function without over-amplifications to enhance the behavior of the component-wise WENO-ACM method in eliminating or significantly reducing the numerical oscillations with improved computational efficiency. Extensive numerical experiments governed by 1D and 2D Euler systems are performed to demonstrate the enhanced performances of the proposed method. We conclude that the proposed method can achieve the formal convergence rates of accuracy in regions with smooth solutions regardless of critical points and it is much more stable than the component-wise WENO-ACM method. Moreover, compared to the characteristic-wise WENO-ACM method, the new method can save 25–57% computational cost. [ABSTRACT FROM AUTHOR]

We present a fast and splitting-based numerical scheme that employs an interpolation method for the system of the reaction-diffusion equations. Typically, the time step restriction arises to the nonlinear reaction terms when we calculate the highly stiff system of reaction-diffusion equations. This issue can be resolved through various implicit solvers, but they shall present another problem of having a longer computing time for each step. In order to overcome these shortcomings, we present a splitting-based hybrid scheme with a pre-iteration process before the main loop to derive interpolating points which are employed to evaluate the intermediate solution, instead of computing the nonlinear reaction term directly. The stability and convergence analysis are provided for selected reaction-diffusion models. We verify that the results of our proposed method are in good agreements with those in the references, as demonstrated numerically. Furthermore, we examine and compare the computing time performance among the methods, and draw that our proposed method yields good results. [ABSTRACT FROM AUTHOR]

In this paper, the space–time fractional Bloch–Torrey equation is discretized in space using second and fourth-order schemes. Then, it leads to a system of Caputo fractional ordinary differential equations over time. A highly accurate exponential integrator is utilized to solve systems of fractional differential equations. We rigorously demonstrate the stability and convergence of the schemes by employing the energy technique. Finally, some numerical examples are presented to validate the accuracy and efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The weak maximum principle of finite element methods for parabolic equations is proved for both semi-discretization in space and fully discrete methods with k-step backward differentiation formulae for k = 1 , ... , 6 , on a two-dimensional general polygonal domain or a three-dimensional convex polyhedral domain. The semi-discrete result is established via a dyadic decomposition argument and local energy estimates in which the nonsmoothness of the domain can be handled. The fully discrete result for multistep backward differentiation formulae is proved by utilizing the solution representation via the discrete Laplace transform and the resolvent estimates, which are inspired by the analysis of convolutional quadrature for parabolic and fractional-order partial differential equations. [ABSTRACT FROM AUTHOR]

The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonian. However in many cases, the structure of the underlying optimal problem leads to a convex but nondifferentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusion (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniqueness of weak solutions to the resulting MFG PDI system under standard assumptions in the literature. We propose a monotone stabilized finite element discretization of the problem, using conforming affine elements in space and an implicit Euler discretization in time with mass-lumping. We prove the strong convergence in L 2 (H 1) of the value function approximations, and strong convergence in L p (L 2) of the density function approximations, together with strong L 2 -convergence of the value function approximations at the initial time. [ABSTRACT FROM AUTHOR]

This paper primarily focuses on numerical threshold stability of a nonlinear diffusive age-structured HIV model. A semi-discrete system based on the line method is established and a threshold is introduced in the stability analysis of the resulting discrete system, which we refer to as numerical basic numerical reproduction number. A discretization technique is adopted in formulating a full-discrete numerical scheme to ensure the biological significance can be preserved without stepsize restriction. It is proved that numerical stability thresholds we proposed converge to the exact stability threshold and replicate the monotonicity and qualitative properties of the latter. Finally, the conclusions are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

This article examines the convergence of the two-step backward difference formula (BDF2)-discontinuous Galerkin (DG) scheme in the context of a nonlocal reaction–diffusion equation. Our methodology integrates BDF2 for temporal discretization with a DG finite element method for spatial discretization, enabling a comprehensive analysis of the nonlocal diffusion equation. We establish optimal-order convergence rates for the fully discrete system. To assess the effectiveness and precision of the proposed scheme, we provide numerical simulations alongside convergence results. [ABSTRACT FROM AUTHOR]

This article intends to contribute a time-dependent mesh method to analyze the singularly perturbed delay parabolic partial differential equation exhibiting the boundary layer instabilities. In engineering control systems, past population effects in ecological models, stress responses in viscoelastic materials, and past pressures or velocities in complex fluid flows are all modeled by delayed partial differential equations. In the vicinity of the boundary layer, where the velocity field exerts a major influence, the solution exhibits sharp gradients. Because these gradients are steep, to obtain accurate solution a numerical method with dynamic mesh is obliged. To compute these singularities more accurately, we imposed adaptive moving mesh method that is a time-dependent self-adaptive method by imposing the center differences in the spatial direction and implicit Euler in the temporal direction. A stability analysis for the proposed scheme is performed using the energy technique. Numerical experiments are performed with various examples, and the approximated results support theoretical findings. Six numerical examples are presented with their errors and convergence orders. [ABSTRACT FROM AUTHOR]

In this article, we consider the Chew, Goldberger and Low (CGL) plasma flow equations, which is a set of nonlinear, non-conservative hyperbolic PDEs modeling anisotropic plasma flows. These equations incorporate the double adiabatic approximation for the evolution of the pressure, making them very valuable for plasma physics, space physics, and astrophysical applications. We first present the entropy analysis for the weak solutions. We then propose entropy-stable finite-difference schemes for the CGL equations. The key idea is to reformulate the CGL equations by rewriting some of the conservative terms in the non-conservation form. The conservative part of the reformulated equations is very similar to the magnetohydrodynamics (MHD) equations which is then symmetrized using Godunov’s symmetrization process for the MHD equations. The resulting equations are in the form where the conservative part combined with non-conservative Godunov’s terms is compatible with the entropy equation and the rest of the non-conservative terms do not contribute to the entropy equations. The final set of reformulated equations is then discretized by designing entropy conservative numerical flux and entropy diffusion operator based on the entropy scaled eigenvectors of the conservative part. We then prove the semi-discrete entropy stability of the schemes for the reformulated CGL equations. The schemes are then tested using several test problems derived from the corresponding MHD test cases. [ABSTRACT FROM AUTHOR]

This paper focuses on the unconditionally optimal error estimates of a fully discrete decoupled scheme for two-phase magnetohydrodynamic (MHD) model with different viscosities and electric conductivities, by using the zero-energy-contribution (ZEC) method for the temporal discretization and mixed finite elements for the spatial discretization. Based on the ZEC property of the nonlinear and coupled terms of the model, an ordinary differential equation is designed to introduce a nonlocal scalar auxiliary variable which will play a key role in the design and the energy stability of the decoupled scheme. Combining fully explicit treatment on the nonlinear and coupled terms with the stabilization method for nonlinear potential, a decoupled temporal discrete scheme is proposed. Utilizing mixed finite elements for the spatial discretization in this temporal discrete scheme, a fully discrete scheme is proposed. Both schemes are proved to be mass-conservative and unconditionally energy stable. The unconditionally optimal error estimates of the temporal discrete scheme are derived for two dimensional and three dimensional (2D/3D) cases. Utilizing a modified Maxwell projection with variable electric conductivities, the superconvergence of its negative norm estimates, mathematical induction, and the unconditional stability of the numerical scheme, we also derive the optimal error estimates in L 2 -norm for the fully discrete scheme in 2D/3D cases, without any restriction on the time step size and mesh size. Finally, numerical experiments are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The variable-step deferred correction method constructs high-order schemes by modifying the low-order schemes, which not only relaxes the strict step-ratio constraints related to higher-order schemes, but also more accurately captures multi-scale evolution processes compared to lower-order schemes. It also provides internal error estimators to adjust the step sizes in adaptive time-stepping algorithms. Two third-order deferred correction methods based on the variable-step second-order BDF formula are analyzed for the Cahn–Hilliard model. The stability of the proposed variable-step deferred correction schemes is established under the step-ratio restriction of 0 < r k < 4.864 . By utilizing the discrete orthogonal convolution kernels and some discrete convolution embedding inequalities, the modified energy dissipation law and the L 2 norm error estimate at the discrete levels are established. Numerical experiments validate the effectiveness of our approaches. [ABSTRACT FROM AUTHOR]

In this research, we present an advanced numerical technique developed by adhering to the consistency condition, leveraging the characteristics of polynomial cubic spline approximations. Our method is specifically tailored for tackling nonlinear equations like the Kuramoto-Sivashinsky and Korteweg-de Vries equations. Unlike conventional finite difference methods, which encounter difficulties in addressing singular biharmonic problems directly, our approach overcomes this hurdle. By incorporating off-step points in the spatial dimension, our method facilitates the direct resolution of singular problems. We assess stability using von Neumann stability analysis on a linear model. Chaotic simulations of the nonlinear Kuramoto–Sivashinsky equation are conducted across various time stages. Single-soliton of the Korteweg-de Vries equation is solved numerically at different time levels. Our proposed technique exhibits high accuracy, demonstrating its efficacy compared to prior research. Additionally, we delve into solving diverse benchmark problems numerically, evaluating global relative and maximum absolute errors in comparison with previous studies. [ABSTRACT FROM AUTHOR]

The fractional KdV equation has significant physical implications, the study of its efficient method has great scientific significance. A fast compact difference (FCD) scheme on graded meshes is used to address the time-fractional nonlinear KdV equation with an initial singularity. The temporal discretization employs the fast L1 algorithm, while a fourth-order compact scheme is utilized for spatial discretization. The Newton's method is used to approximate the nonlinear term. The fast L1 algorithm can greatly shorten the CPU calculation time without losing the precision of numerical solution. We verify the existence and uniqueness of numerical solutions, while also provide unconditional stability and convergence analysis. Theoretical analysis is validated by numerical results, indicating the FCD scheme on graded meshes has high-accuracy and time-saving performance. [ABSTRACT FROM AUTHOR]

Objective: In this work, singularly perturbed time dependent delay parabolic convection-diffusion problem with Dirichlet boundary conditions is considered. The solution of this problem exhibits boundary layer at the right of special domain. In this layer the solution experiences steep gradients or oscillation so that traditional numerical methods may fail to provide smooth solutions. We developed oscillation free parameter uniform exponentially spline numerical method to solve the considered problem. Results: In the temporal direction, the implicit Euler method is applied, and in the spatial direction, an exponential spline method with uniform mesh is applied. To handle the effect of perturbation parameter, an exponential fitting factor is introduced. For the developed numerical scheme, stability and uniform error estimates are examined. It is shown that the scheme is uniformly convergent of linear order in the maximum norm. Numerical examples are provided to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

The time-fractional fourth-order reaction-diffusion problem, which contains more than one time-fractional derivative of orders lying between 0 and 1, is considered. This problem is the generalized version of the problem discussed by Nikan et al. Appl. Math. Model. 89 (2021), 819–836 that has only one time-fractional derivative. It is widely used in the study of chemical waves and patterns in reaction-diffusion systems. The analysis of non-smooth solutions to this problem is discussed broadly using the Caputo-time fractional derivative. The non-smooth solutions to the problem have a weak singularity close to zero that can be efficiently handled by considering the non-uniform mesh. The method based on the non-uniform time stepping is an efficacious way to regain accuracy. The current study presents the trigonometric quintic B-spline approach to solve this multi-term time-fractional fourth-order problem using graded mesh and effective grading parameters. The stability and convergence results are proved through rigorous analysis, which helps choose the optimal grading parameter. The accuracy and effectiveness of our technique are observed in our numerical experiments that manifest the comparison of uniform and non-uniform meshes. [ABSTRACT FROM AUTHOR]

This paper primarily lies in presenting an α -robust analysis of finite element method for space-time fractional diffusion equation. To this end, we firstly develop finite element approximation to fractional Laplacian and provide a d-dimensional fast Fourier transform (FFT)-based fast algorithm to derive spatial discretization of space-time fractional diffusion problem. Then we study the α -robust stability and error analyses of full discrete scheme of space-time fractional diffusion problem using L1 scheme on graded temporal meshes. Different from current many existing works, the established stability and error bounds will not blow-up under energy norm as α → 1 - , and the present error analysis illustrates that a choice of time mesh graded factor γ = (2 - α) / α shall yield an optimal rate of convergence O (N - (2 - α)) in temporal direction, where N is the number of temporal meshes. Eventually, some numerical tests are given to show the efficiency and α -robust behavior of the proposed scheme. [ABSTRACT FROM AUTHOR]

We prove the stability of entropy solutions of nonlinear conservation laws with respect to perturbations of the initial datum, the space-time dependent flux and the entropy inequalities. Such a general stability theorem is motivated by the study of problems in which the flux P[u](t, x, u) depends possibly non-locally on the solution itself. For these problems we show the conditional existence and uniqueness of entropy solutions. Moreover, the relaxation of the entropy inequality allows to treat approximate solutions arising from various numerical schemes. This can be used to derive the rate of convergence of the recent particle method introduced in Radici and Stra (SIAM J Math Anal 55(3):2001–2041, 2023. https://doi.org/10.1137/21M1462994) to solve a one-dimensional model of traffic with congestion, as well as recover already known rates for some other approximation methods. [ABSTRACT FROM AUTHOR]

Furihata and Matsuo proposed in 2010 an energy-conserving scheme for the Zakharov equations, as an application of the discrete variational derivative method (DVDM). This scheme is distinguished from conventional methods (in particular the one devised by Glassey (Math Comput 58(197):83–102, 1992)) in that the invariants are consistent with respect to time, but it has not been sufficiently studied both theoretically and numerically. In this study, we theoretically prove the solvability under the loosest possible assumptions. We also prove the convergence of this DVDM scheme by improving the argument by Glassey. Furthermore, we perform intensive numerical experiments for comparing the above two schemes. It is found that the DVDM scheme is superior in terms of accuracy, but since it is fully-implicit, the linearly-implicit Glassey scheme is better for practical efficiency. In addition, we proposed a way to choose a solution for the first step that would allow Glassey's scheme to work more efficiently. [ABSTRACT FROM AUTHOR]

The target of this study is a norm-conservative scheme for the Ostrovsky equation, as its mathematical analysis has not been addressed. First, the existence and uniqueness of its numerical solutions are demonstrated. Subsequently, a convergence estimate in the two-norm is established. This, in turn, implies a convergence in the first-order Sobolev space using a supplementary sup-norm boundedness argument. Finally, this conservative scheme can be implemented in a differential form, which is considerably better than the integral form in terms of computational cost-effectiveness. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) has been widely used as a spatial discretization method for phase field models since the seminal work of Gómez et al. (Comput. Methods Appl. Mech. Engrg. 197(49), pp. 4333–4352, 2008), and the first numerical convergence study of IGA for the Cahn-Hilliard equation was presented by Kästner et al. (J. Comput. Phys. 305(15), pp. 360–371, 2016). However, to the best of our knowledge, the theoretical convergence analysis of IGA for the Cahn-Hilliard equation is still missing in the literature. In this paper, we provide the convergence analysis of IGA for the multi-dimensional Cahn-Hilliard equation for the first time. The two important steps to carry out the convergence analysis are (1) we rigorously prove that the L ∞ norm of IGA solution is uniformly bounded for all mesh sizes, and (2) we construct an appropriate Ritz projection operator for the bi-Laplacian term in the Cahn-Hilliard equation. The first- and second-order stabilized semi-implicit schemes are used to obtain the fully discrete schemes. The energy stability analyses are rigorously proved for the resulting fully discrete schemes. Finally, several two- and three-dimensional numerical examples are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

A Lawson-type exponential integrator combined with the Fourier pseudo-spectral method is provided for the nonlinear Double Sine-Gordon equation (DSGE), while the nonlinearity is characterized by β / ϵ with small parameter ϵ ∈ (0 , 1 ] and interaction parameter β ∈ (0 , + ∞) . In comparison to the Sine-Gordon equation, DSGE has many properties of solitons as well as its own unique new features. This is the first work to numerically simulate the physical phenomena arising from DSG kinks collisions. The improved uniform error bounds are proved by using the regularity compensation oscillatory (RCO) technique, which are O (α 2 τ + h m) up to the long time at T ϵ = T / α 2 , where α = ϵ for β ≥ 1 and α = ϵ / β for 0 < ϵ < β < 1 . Based on the uniform bounds, the error estimation for the discrete energy is derived. Furthermore, the improved error bounds are extended to two oscillatory DSGEs with O (ϵ 2) and O (ϵ 2 / β 2) wavelength in time. Numerical examples are provided to illustrate the accuracy and discrete energy property of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new semi-Lagrangian numerical method proposed to solve a cell population balance model which describes cell dwarfism, by allowing cell division at any size. We analyse its convergence and derive an optimal rate. Numerical experiments are reported to demonstrate the predicted accuracy of the scheme. Finally, the good behaviour of the numerical method is exhibited in stressed conditions that can provoke lack of smoothness and simulations are included to show how the increase in the division rate of small size-cells promotes dynamics in which nonfunctional dwarf cells are saturating the total cell population. [ABSTRACT FROM AUTHOR]

This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes operator become non-coercive and indefinite, and under the smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes projection to deal with the Stokes part and, with the help of a special projection, to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed. [ABSTRACT FROM AUTHOR]

This report investigates a stabilization method for first order hyperbolic differential equations applied to DNA transcription modeling. It is known that the usual unstabilized finite element method contains spurious oscillations for nonsmooth solutions. To stabilize the finite element method the authors consider adding to the first order hyperbolic differential system a stabilization term in space and time filtering. Numerical analysis of the stabilized finite element algorithms and computations describing a few biological settings are studied herein. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an operator-splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Lévy noise. More specifically, using a variant of the classical Kružkov doubling of variables approach, we show that the approximate solutions generated by the splitting scheme converge to the unique stochastic entropy solution of the underlying problems. Finally, the convergence analysis is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

An artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find initial values, $ [u(.,0), v(.,0), w(.,0)] $ [ u (. , 0) , v (. , 0) , w (. , 0) ] , that can evolve into a useful approximation to a hypothetical target result $ [u^*(.,T_{\max }), v^*(.,T_{\max }), w^*(T_{\max })] $ [ u ∗ (. , T max) , v ∗ (. , T max) , w ∗ (T max) ] , at some realistic $ T_{\max } > 0 $ T max > 0. Highly non smooth target data are considered, that may not correspond to actual solutions at time $ T_{\max } $ T max . Stabilization is achieved by applying a compensating smoothing operator at each time step. Such smoothing leads to a distortion away from the true solution, but that distortion is small enough to allow for useful results. Data assimilation is illustrated using $ 512 \times 512 $ 512 × 512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Computational experiments show that efficient FFT-synthesized smoothing operators, based on $ (-\Delta)^q $ (− Δ) q with real q>3, can be successfully applied, even in nonlinear problems in non-rectangular domains. However, an example of failure illustrates the limitations of the method in problems where $ T_{\max } $ T max , and/or the nonlinearity, are not sufficiently small. [ABSTRACT FROM AUTHOR]

The objective of this work is to provide an efficient numerical scheme for solving singularly perturbed parabolic convection-diffusion problems with a large time lag having Robin-type boundary conditions. The implicit Euler scheme is used in the temporal direction on a uniform mesh. To handle the layer behavior caused due to the presence of perturbation parameter, the problem is solved using the upwind scheme on two layer-resolving meshes in the spatial direction. As the Shishkin mesh makes the order of convergence to one up to a logarithmic factor, in the spatial direction, the presence of this effect is prevented by the use of the Bakhvalov-Shishkin mesh. Further, the robustness of the scheme is tested over semi-linear time-lagged initial boundary value problems. Numerical outputs are presented in the form of tables and graphs to prove the parameter-uniform nature and robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods. [ABSTRACT FROM AUTHOR]

In this paper, we develop a stabilized Gauge-Uzawa discontinuous Galerkin (DG) method with second-order time-accurate for the incompressible magnetohydrodynamic equations. The proposed scheme here possess the properties of the linearity, fully decoupling and unconditional energy stability, which is employed by combining a second-order projection-type Gauge-Uzawa method for the Navier–Stokes equations, a stabilization strategy for Maxwell’s equations, the implicit-explicit frameworks for the nonlinear coupling terms and the interior penalty DG method for the spatial approximation. We rigorously prove the unique solvability, unconditional energy stability and optimal error estimates of the proposed scheme. Finally, several numerical examples are provided to demonstrate the accuracy, stability, and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of high-order, large time-stepping, and delay-free stabilization schemes for the Allen-Cahn equation. First, we apply a Fourier pseudo-spectral method for spatial discretization, and then, we establish the l 2 -bound of the semi-discrete system. Furthermore, by adopting a time-step-dependent stabilization technique and taking advantage of recursive approximation of the exponential functions, we propose a class of stabilization Runge-Kutta schemes that preserve l 2 -bound for any time-step size. Finally, we eliminate the delayed convergence brought by stabilization via a relaxation technique. Consequently, the resulting up-to-fourth-order parametric relaxation integrating factor Runge-Kutta (pRIFRK) schemes preserve the l 2 -boundedness unconditionally with suitably chosen stabilization parameters. We also prove that the first-order pRIFRK scheme is unconditionally dissipative, w.r.t. a modified energy function, and the temporal convergence in the l 2 -norm is estimated with pth-order accuracy. Numerical experiments are carried out to demonstrate the high-order accuracy, structure-preserving properties, and performance of the proposed schemes. [ABSTRACT FROM AUTHOR]

We study explicit strong stability preserving (SSP) multiderivative general linear methods (MDGLMs) for the numerical solution of hyperbolic conservation laws. Sufficient conditions for MDGLMs up to four derivatives to be SSP are determined. In this work, we describe the construction of two external stage explicit SSP MDGLMs based on Taylor series conditions, and present examples of constructed methods up to order nine and three internal stages along with their SSP coefficients. It is difficult to apply these methods directly to the discretization of partial differential equations, as higher-order flux derivatives must be calculated analytically. We hence use a Jacobian-free approach based on the recent development of explicit Jacobian-free multistage multiderivative solvers (Chouchoulis et al. J. Sci. Comput. 90, 96, 2022) that provides a practical application of MDGLMs. To show the capability of our novel methods in achieving the predicted order of convergence and preserving required stability properties, several numerical test cases for scalar and systems of equations are provided. [ABSTRACT FROM AUTHOR]

In the present paper, to fill the gap of the effect of singularity arising from multiple fractional derivatives on numerical analysis, the regularity and high order difference scheme for time fractional telegraph equations are taken into consideration. Firstly, the analytic solution is obtained by employing Laplace transform, and its regularity is then deduced. Secondly, by the technic of decomposition, the improved regularity of solution is derived. Furthermore, to overcome the weak singularity and enhance convergence precision, a second-order fitted scheme based on L2- 1 σ approximation and order reduction method is applied to such problems, which is an improvement for the work [6]. Ultimately, examples are presented to verify the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper. [ABSTRACT FROM AUTHOR]