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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]

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]

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]

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]

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]

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]

In this paper, we establish a unified analysis of both the conforming and nonconforming virtual element methods for a class of nonlinear Sobolev equations in which the nonlinear diffusion coefficients, a(u) and b(u), may not satisfy global Lipschitz continuity or elliptical conditions. The discretization for the spatial variables is the virtual element method on general polygonal meshes, whereas the temporal discretization is accomplished with a linearized Crank–Nicolson scheme. By employing a temporal-spatial error splitting argument and introducing a novel projection operator, we prove that the numerical solution is bounded in the L ∞ -norm without any time step-size conditions. Moreover, for both conforming and nonconforming ones, optimal error estimates in L 2 -norm and H 1 -seminorm for the fully scheme are obtained unconditionally. Numerical results are provided on different polygonal meshes to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose several linear fully discrete schemes for the mass-conserved Allen-Cahn-Navier–Stokes equation, based on the generalized stabilized exponential scalar auxiliary variable approach in time and the marker and cell (MAC) scheme in space. It is quite remarkable that our schemes can guarantee second-order accuracy in space provided the maximum bound principle (MBP) is satisfied, whereas most previous work can only possess first-order accuracy in space. We rigorously show that the constructed schemes satisfy the unconditional energy dissipation law and preserve the MBP. Finally, various numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish improved uniform error bounds under H α / 2 -norm (1 < α ≤ 2) for the long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation (NSFSGE) by a Lawson-type exponential integrator Fourier pseudo-spectral (LEI-FP) method. Firstly, a Lawson-type exponential integrator method is used to discretize the time direction. Then, the Fourier pseudo-spectral method is applied to discretize the space direction. We rigorously prove that the equation is energy conservation in a continuous state. Regularity compensation oscillation (RCO) technique is employed to strictly prove the improved uniform error bounds at O ε 2 τ in temporal semi-discretization and O h m + ε 2 τ in full-discretization up to the long-time T ε = T / ε 2 ( T > 0 fixed), respectively. To obtain the convergence order h m in space, we only need to directly prove it instead of proving that the numerical solution is H m + α / 2 -norm bounded as before. Complex NSFSGE and oscillatory NSFSGE are also discussed. This is the novel work to construct the improved uniform error bounds for the long-time dynamics of the high-dimensional nonlinear space fractional Klein-Gordon equation with non-polynomial nonlinearity. Finally, numerical examples in two-dimension and three-dimension are provided to confirm the improved error bounds, and we find drastically different evolving patterns between NSFSGE and the classical sine-Gordon equation. [ABSTRACT FROM AUTHOR]

In this paper we develop the numerical analysis for a non-isothermal diffuse-interface model, in dimension N = 2 , 3 , that describes the movement of a mixture of two incompressible viscous fluids. This model consists of modified Navier–Stokes equations coupled with a phase-field equation given by a convective Allen–Cahn equation, and energy transport equation for the temperature; which admits a dissipative energy inequality. We propose an energy stable numerical scheme based on the Finite Element Method, and we analyze optimal weak and strong error estimates, as well as convergence towards regular solutions. In order to construct the numerical scheme, we introduce two extra variables (given by the gradient of the temperature and the variation of the energy with respect to the phase-field function) which allows us to control the strong regularity required by the model, which is one of the main difficulties appearing from the numerical point of view. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed, energy stable and satisfies a set of uniform estimates which allow to analyze the convergence of the scheme. Finally, we present some numerical simulations to validate numerically our theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a fourth-order numerical scheme for solving multi-asset nonlinear variable coefficient Black–Scholes PDEs. First, using the Newton linearization technique, we linearize the given nonlinear PDE, and obtain a sequence of linear PDEs. Then, we discretize the time derivative by the Crank–Nicolson scheme, and the resultant semi-discrete problem by the central difference scheme on uniform meshes. In order to enhance the order of convergence of the proposed scheme, we use the Richardson extrapolation method, by solving the fully discrete problem on two different meshes. The stability and convergence are studied. To validate the proposed technique, several numerical experiments are carried out. [ABSTRACT FROM AUTHOR]

We propose an alternate overlapping Schwarz method to solve the singularly perturbed problems of convection-diffusion type. The method decomposes the original domain into two overlapping subdomains; one subdomain is known as the outside boundary layer subdomain and the other subdomain is known as the inside boundary layer subdomain, which contains the boundary layer. On the outside boundary layer subdomain, a combination of the compact difference scheme and the central difference scheme with uniform mesh is considered, while on the inside boundary layer subdomain, a combination of the compact difference scheme and the central difference scheme with a special type of piecewise-uniform mesh is considered. The convergence analysis is given and the method is shown to have almost second order global parameter-uniform convergence. The method is then extended to a weakly coupled system of M (≥ 2) singularly perturbed convection-diffusion problems. Numerical experiments are given in support of the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study two discontinuous Galerkin (DG) methods: the Symmetric Interior Penalty Galerkin (SIPG) method and the Non-symmetric Interior Penalty Galerkin (NIPG) method. These methods are applied to nonlinear parabolic delay problems of nonmonotone type considering the locally Lipschitz type nonlinearity and using the backward finite difference method for time discretization. By employing Brouwer’s fixed point theorem and by showing Lipschitz’s continuity of a discrete map, we first prove the existence of a discrete solution and then establish the uniqueness of the discrete solution. After this, we derive the stability of the backward scheme in L 2 norm and energy norm. We also discuss error bounds for both versions of DG methods (SIPG and NIPG) under various hypotheses on the regularity of the solution. Finally, we validate our theoretical results by finding numerical solutions for different population models. We also study the behavior of the matured population for different initial functions in the Nicholson Blowflies equation with time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical approximations of a coupled nonlinear Schrödinger equation, which describes the nonlinear dynamics process of protein folding. The main numerical challenge in solving this system is how to discretize nonlinear terms to ensure unconditional energy stability at the discrete level. In order to address this numerical problem, we construct a linear, decoupled, and second-order time-accurate numerical scheme, which is employed by the scalar auxiliary variable approach for the nonlinear terms combined with the finite element method (FEM) for spatial discretization, and the implicit–explicit approach for the highly nonlinear and coupling terms. The unconditional energy stability and unique solvability of the fully discrete scheme are rigorously proved. Especially, using the temporal-spatial error splitting argument, we obtain optimal L 2 -error estimates for r-order FEM overcoming time-step restriction. Finally, numerical results are provided to illustrate the theoretical predictions and demonstrate the efficiency of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze new first- and second-order implicit-explicit (IMEX) schemes for the unsteady Navier–Stokes-Darcy model to describe the coupled free flow-porous media system, which is based on the scalar auxiliary variable (SAV) approach in time and finite element method in space. The constructed schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and can decouple the Navier–Stokes and Darcy systems. The unconditional stability of both the first- and second-order IMEX schemes can be derived for the coupled system equipped with the Lions interface condition, where the key point is that we should construct a new trilinear form to balance the fully explicit discretizations of the nonlinear terms in the complex system. We can also establish rigorous error estimates for the velocity and hydraulic head of the first-order scheme without any time step restriction. Numerical examples are presented to validate the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper a time-fractional Black–Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and 2 - μ is time, where μ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option. Moreover, the classical Black–Scholes model is also incorporated into our numerical study to validate the competence of our method in handling not only fractional problems, but also classical ones with favorable results. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments. [ABSTRACT FROM AUTHOR]

We present a parameter-robust finite difference method for solving a system of weakly coupled singularly perturbed convection-diffusion equations. The diffusion coefficient of each equation is a small distinct positive parameter. Due to this, the solution to the system has, in general, overlapping boundary layers. The problem is discretized using a particular combination of a compact second-order difference scheme and a central difference scheme on a piecewise-uniform Shishkin mesh. The convergence analysis is given, and the method is shown to have almost second-order uniform convergence in the maximum norm with respect to the perturbation parameters. The results of numerical experiments are in agreement with the theoretical outcomes. [ABSTRACT FROM AUTHOR]

This work considers the numerical analysis of nonlinear Volterra integrodifferential equations that arise from, e.g., the theory of isotropic viscoelastic rods and plates. Novel properties of the kernel and its derivatives are derived via the Laplace transform, and the regularity of the solutions is proved. Then we apply the Crank–Nicolson method and the second-order convolution quadrature rule to develop the discrete-in-time scheme, and the energy argument is employed to analyze the stability and convergence of the proposed scheme. Numerical experiments are performed to substantiate the theoretical findings. [ABSTRACT FROM AUTHOR]

A shallow-water wave propagation model can be described as a generalized Rosenau–Burgers equation with strong nonlinearity and high-order dispersion terms. In this paper, we propose two generalized high-order (up to eighth-order) compact finite difference schemes for solving the generalized Rosenau–Burgers equation. The first scheme is a two-level nonlinear Crank–Nicolson difference scheme and the second is a three-level linearized difference scheme. We derive the discrete mass and energy properties, and provide rigorous proofs for the boundedness, existence, and convergence with order O (τ 2 + h s) (s = 4 , 6 , 8) of these proposed generalized compact difference schemes, where τ and h denote the time- and space-steps, respectively. Finally, the validity of the theoretical analysis is verified through numerical experiments, confirming the effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this article, we present a fully discrete Crank–Nicolson Galerkin finite element method for solving the two-dimensional nonlinear extended-Fisher–Kolmogorov equation: u t + γ Δ 2 u - Δ u - u + u 3 = 0. The boundedness of the numerical solution in the maximum norm, unique solvability, and related convergence results in L 2 and L ∞ -norms are studied in detail. Also, a new linearized Crank–Nicolson Galerkin modification scheme is designed and error estimate without any time step restrictions is established. Finally, some computational experiments in one and two dimension cases are provided to illustrate the efficacy of our method and to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present the L 2 -norm stability analysis and error estimate for the explicit single-step time-marching discontinuous Galerkin (DG) methods with stage-dependent numerical flux parameters, when solving a linear constant-coefficient hyperbolic equation in one dimension. Two well-known examples of this method include the Runge–Kutta DG method with the downwind treatment for the negative time marching coefficients, as well as the Lax–Wendroff DG method with arbitrary numerical flux parameters to deal with the auxiliary variables. The stability analysis framework is an extension and an application of the matrix transferring process based on the temporal differences of stage solutions, and a new concept, named as the averaged numerical flux parameter, is proposed to reveal the essential upwind mechanism in the fully discrete status. Distinguished from the traditional analysis, we have to present a novel way to obtain the optimal error estimate in both space and time. The main tool is a series of space–time approximation functions for a given spatial function, which preserve the local structure of the fully discrete schemes and the balance of exact evolution under the control of the partial differential equation. Finally some numerical experiments are given to validate the theoretical results proposed in this paper. [ABSTRACT FROM AUTHOR]

We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is O (ε) with 0 < ε ≪ 1 a dimensionless parameter up to the time at O (1 / ε 2) . The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discretization at O (ε 2 τ) instead of O (τ) according to classical error estimates and at O (h m + ε 2 τ) for the full-discretization up to the time T ε = T / ε 2 with T > 0 fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the nonlinear Klein–Gordon equation with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with O (ε 2) wavelength in time and O (ε - 2) wave speed, which indicates that the temporal error is independent of ε when the time step size is chosen as O (ε 2) . Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp. [ABSTRACT FROM AUTHOR]

In this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and H (div ; Ω) -conforming space of order k + 1 , respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete H 1 -stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the L 2 projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete H 1 -norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete H 1 -norm is achieved. Several numerical experiments are carried out to confirm the proposed theories. [ABSTRACT FROM AUTHOR]

We propose second-order numerical methods based on the generalized positive auxiliary variable (gPAV) framework for solving the Cahn–Hilliard–Navier–Stokes–Darcy model in superposed free flow and porous media. In the gPAV-reformulated system, we introduce an auxiliary variable according to the modified energy law and take account into the interface conditions between the two subdomains. By implicit-explicit temporal discretization, we develop fully decoupled linear gPAV-CNLF and gPAV-BDF2 numerical methods effected with the Galerkin finite element method. The fully discrete schemes satisfy a modified energy law irrespective of time step size. Plentiful numerical experiments are performed to validate the methods and demonstrate the robustness. The application in filtration systems, the influence of viscous instability, general permeability, curve interface, and different densities are discussed in details to further illustrate the compatibility and applicability of our developed gPAV numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, by introducing two temporal derivative-dependent auxiliary variables, a linearized and decoupled fourth-order compact finite difference method is developed and analyzed for the nonlinear coupled bacterial systems. The temporal-spatial error splitting technique and discrete energy method are employed to prove the unconditional stability and convergence of the method in discrete maximum-norm. Furthermore, to improve the computational efficiency, an alternating direction implicit (ADI) compact difference algorithm is proposed, and the unconditional stability and optimal-order maximum-norm error estimate for the ADI scheme are also strictly established. Finally, several numerical experiments are conducted to validate the theoretical convergence and to simulate the phenomena of bacterial extinction as well as the formation of endemic diseases. In particular, an adaptive time-stepping algorithm is developed and tested for long-term stable simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a time two-grid algorithm to get the numerical solution of nonlinear generalized viscous Burgers' equation, which is the first time that the two-grid method is used to solve this problem. Based on Crank–Nicolson finite difference scheme, we establish the time two-grid difference (TTGD) scheme which consists of three computational procedures to reduce the computational cost compared with the general finite difference (GFD) scheme. The cut-offf function method is applied to prove the conservation, unique solvability, the prior estimate and convergence in L 2 -norm and L ∞ -norm of the TTGD scheme on the coarse grid and fine grid, respectively. Comparing our TTGD scheme with GFD scheme in Zhang et al. (Appl Math Lett 112:106719, 2021), we provided the proof the uniqueness the solution of the nonlinear scheme, direct proof of convergence in L 2 -norm and the prior estimate both on the coarse mesh and fine mesh. The numerical results show that our TTGD scheme is more efficient than the GDF scheme in Zhang et al. (2021) in terms of the CPU time. Particularly, our method not only improves the efficiency, but also preserves the energy conservation of the original model. [ABSTRACT FROM AUTHOR]

Fractional initial-value problems (IVPs) and time-fractional initial-boundary value problems (IBVPs), each with a Caputo temporal derivative of order α ∈ (0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O(N−2) convergent for IVPs on suitably graded meshes with N points, thereby improving the O(N−(2−α)) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O(N−2) convergence in time, together with O(h2) convergence in space where h is the spatial mesh width. Numerical experiments support our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical approximations of the Cahn-Hilliard-Darcy-Stokes system, which is a combination of the modified Cahn-Hilliard equation with the Darcy-Stokes equation. A novel discontinuous Galerkin pressure-correction scheme is proposed for solving the coupled system, which can achieve the desired level of linear, fully decoupled, and unconditionally energy stable. The developed scheme here is implemented by combining several effective techniques, including by adding an additional stabilization term artificially in Cahn-Hilliard equation for balancing the explicit treatment of the coupling term, the stabilizing strategy for the nonlinear energy potential, and a rotational pressure-correction scheme for the Darcy-Stokes equation. We rigorously prove the unique solvability, unconditional energy stability, and optimal error estimates of the proposed scheme. Finally, a number of numerical examples are provided to demonstrate numerically the efficiency of the present formulation. [ABSTRACT FROM AUTHOR]

In this paper, a linearized structure-preserving Galerkin finite element method is investigated for Poisson-Nernst-Planck (PNP) equations. By making full use of the high accuracy estimation of the bilinear element, the mean value technique and rigorously dealing with the coupled nonlinear term, not only the unconditionally optimal error estimate in L 2 -norm but also the unconditionally superclose error estimate in H 1 -norm for the related variables are obtained. Then, the unconditionally global superconvergence error estimate in H 1 -norm is derived by a simple and efficient interpolation post-processing approach, without any coupling restriction condition between the time step size and the space mesh width. Finally, numerical results are provided to confirm the theoretical findings. The numerical scheme preserves the global mass conservation and the electric energy decay, and this work has a great improvement of the error estimate results given in Prohl and Schmuck (Numer. Math. 111, 591–630 2009) and Gao and He (J. Sci. Comput. 72, 1269–1289 2017). [ABSTRACT FROM AUTHOR]

In this work, we construct a fully discrete scheme with finite difference method based on the staggered grids for the viscous Cahn–Hilliard equation. The constructed scheme can satisfy the unconditional dissipation law with original energy. We carry out a rigorous error analysis with superconvergence by introducing an auxiliary function depending on the chemical potential. We obtain second order accuracy in both space and time with l ∞ (0 , T ; H 1 (Ω)) norm for the phase function and l 2 (0 , T ; l 2 (Ω)) for the chemical potential on non-uniform grids. Numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on modeling and analyzing a nonlinear fractional Maxwell fluid across a vertical plate using a novel definition of the Caputo fractional derivative. Specifically, the newly formulated fractional definition is tailored for the finite difference method. Buoyancy effects are considered to bring the model closer to the natural occurrences of the magnetic field. The novel fractional derivative definition of the designed model has been implemented, and the results have been further analyzed using theoretical and numerical methods. The range of fractional parameters α and β is 0 < α ≤ 1 and 0 < β ≤ 1 , respectively. Significant theorems are also included in the study. A simulation was performed to determine which fractional derivative operator is more accurate than the others. The designed definition demonstrates a high convergence rate and decreased computational cost, both of which are primary objectives. Graphs have also been plotted to illustrate the velocity and temperature distributions as the physical parameters vary. [ABSTRACT FROM AUTHOR]

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/"on top" of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders. [ABSTRACT FROM AUTHOR]

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/"on top" of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43–77, 1991; SIAM J Numer Anal 32(3):706–740, 1995). The greatest modification is the introduction of a Ritz-like "shift operator" that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders. [ABSTRACT FROM AUTHOR]

We propose and analyze a mixed finite element method for the spatial approximation of a time-fractional Fokker–Planck equation in a convex polyhedral domain, where the given driving force is a function of space. Taking into account the limited smoothing properties of the model, and considering an appropriate splitting of the errors, we employed a sequence of clever energy arguments to show optimal convergence rates with respect to both approximation properties and regularity results. In particular, error bounds for both primary and secondary variables are derived in L 2 -norm for cases with smooth and nonsmooth initial data. We further investigate a fully implicit time-stepping scheme based on a convolution quadrature in time generated by the backward Euler method. Our main result provides pointwise-in-time optimal L 2 -error estimates for the primary variable. Numerical examples are then presented to illustrate the theoretical contributions. [ABSTRACT FROM AUTHOR]

In this paper, we describe a new nonlinear finite-volume scheme that preserves the discrete maximum principle (DMP) for the two-dimensional sub-diffusion equation on distorted meshes. One distinguishing feature of our method is its ability to uphold the DMP for the anisotropic sub-diffusion problems, thereby ensuring the absence of spurious oscillations in numerical solutions and maintaining the physical bounds of various quantities, such as concentration, temperature, and density. Notably, our scheme offers the advantage of being applicable to distorted meshes without stringent constraints. Numerical results demonstrate that our scheme successfully preserves maximum principle on various randomly distorted meshes. [ABSTRACT FROM AUTHOR]

This article deals with a coupled system of singularly perturbed convection-diffusion parabolic partial differential equations (PDEs) possessing overlapping boundary layers. As the thickness of the layer shrinks for small diffusion parameter, efficient capturing of the solution and the diffusive flux (i.e., scaled first-order spatial derivative of the solution) leads to a difficult task. It is well-known that the classical numerical techniques have deficiencies in estimating the solution and the diffusive flux on equidistant mesh unless the mesh-size is adequately large. We aim to generate an efficient numerical approximation to the coupled system of PDEs by employing the implicit-Euler method in time and a classical finite difference scheme in space on a layer-adapted Shishkin mesh. Firstly, we discuss about parameter-uniform convergence of the numerical solution in C 0 -norm followed by the error analysis for the scaled discrete space derivative and the discrete time derivative. Subsequently, the parameter-uniform error bound is established in weighted C 1 -norm for global approximation to the solution and the space-time solution derivatives. The theoretical findings are verified by generating the numerical results for two test examples. [ABSTRACT FROM AUTHOR]

A singularly perturbed transport equation is considered. A variable two-step backward differentiation formulas (BDF2) on a Shishkin-type mesh is used to discrete the first-order derivatives of the singularly perturbed transport equation. The stability and error analysis are derived by using the discrete orthogonal convolution kernels. It is proved that the scheme is second-order uniformly convergent with respect to the small parameter, which improves previous results. Numerical experiments are presented to support the theoretical result. [ABSTRACT FROM AUTHOR]

A two-dimensional time-fractional mobile-immobile diffusion problem with the Caputo time-fractional derivative of order α ∈ (0 , 1) is considered. We show that the solution of the problem has a weak singularity at the initial time. Using the averaged L1 formula to approximate the Caputo time-fractional derivative and using a compact finite difference approximation to discretize the space derivatives, we propose a high-order averaged L1-type compact finite difference method on the uniform space-time mesh for the problem. We then base on this method to develop an averaged L1-type compact alternating direction implicit (ADI) finite difference method and a fast sum-of-exponentials compact ADI finite difference method, both of which significantly reduce the storage requirements and the computational costs while maintaining the same global convergence rate. By using the discrete energy analysis technique, we rigorously prove that all methods are unconditionally stable and convergent, and they have the spatial global fourth-order convergence rate and the temporal global convergence rate of order min { 2 , 3 - 2 α } . For the case of α > 1 / 2 , we use the discrete minimum-maximum principle to prove that the temporal second-order convergence rate can also be achieved in positive time. Numerical results confirm the theoretical analysis results and demonstrate the computational efficiency of the methods. [ABSTRACT FROM AUTHOR]

Optimal quadratic spline collocation (QSC) method has been widely used in various problems due to its high-order accuracy, while the corresponding numerical analysis is rarely investigated since, e.g., the perturbation terms result in the asymmetry of optimal QSC discretization. We present numerical analysis for the optimal QSC method in two space dimensions via discretizing a nonlinear time-fractional diffusion equation for demonstration. The L2-1 σ formula on the graded mesh is used to account for the initial solution singularity, leading to an optimal QSC–L2-1 σ scheme where the nonlinear term is treated by the extrapolation. We provide the existence and uniqueness of the numerical solution, as well as the second-order temporal accuracy and fourth-order spatial accuracy with proper grading parameters. Furthermore, we consider the fast implementation based on the sum-of-exponentials technique to reduce the computational cost. Numerical experiments are performed to verify the theoretical analysis and the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation with a defocusing nonlinearity which is mass-(super)critical and energy-subcritical. We prove uniform in time error estimates for the Lie–Trotter time splitting discretization. This uniformity in time is obtained thanks to a vectorfield which provides time decay estimates for the exact and numerical solutions. This vectorfield is classical in scattering theory and requires several technical modifications compared to previous error estimates for splitting methods. [ABSTRACT FROM AUTHOR]

This paper deals with an ε -uniform scheme for delay parabolic singularly perturbed problem (DPSPP). The considered problem is a convection–diffusion (C–D) type with the convection coefficient vanishing inside the domain. The turning point leads to the formation of two exponential boundary layers in the exact solution. We numerically solve the problem using the fitted mesh method and analyze the upwind scheme on a non-uniform mesh. We state some analytical results on the exact solution, which will be required in the convergence analysis of the proposed method. The proposed scheme has order of convergence almost one in space variable and one in time variable. The numerical findings practically support the theoretical results. [ABSTRACT FROM AUTHOR]

A rapidly converging domain decomposition algorithm is introduced for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers. Firstly, a space-time decomposition of the original problem is considered. Subsequently, an iterative process is proposed, wherein the exchange of information to neighboring subdomains is accomplished through the utilization of piecewise-linear interpolation. It is shown that the algorithm provides uniformly convergent numerical approximations to the solution. Our analysis utilizes some novel auxiliary problems, barrier functions, and a new maximum principle result. More importantly, we showed that only one iteration is needed for small values of the perturbation parameter. Some numerical results supporting the theory and demonstrating the effectiveness of the algorithm are presented. [ABSTRACT FROM AUTHOR]

We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrödinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrödinger equation. We use a four term operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schrödinger equation. [ABSTRACT FROM AUTHOR]