High-order Convergence Research Articles

Modified Patankar Linear Multistep Methods for Production-Destruction Systems

Abstract Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the resulting discretizations retain, with no restrictions on the step size, the positivity of the solution and the linear invariant of the continuous-time system. Moreover, we provide results on arbitrarily high order of convergence and we introduce an embedding technique for the Patankar weights denominators to achieve it.

Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

Abstract Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.

Analysis of RL electric circuits modeled by fractional Riccati IVP via Jacobi-Broyden Newton algorithm.

This paper focuses on modeling Resistor-Inductor (RL) electric circuits using a fractional Riccati initial value problem (IVP) framework. Conventional models frequently neglect the complex dynamics and memory effects intrinsic to actual RL circuits. This study aims to develop a more precise representation using a fractional-order Riccati model. We present a Jacobi collocation method combined with the Jacobi-Newton algorithm to address the fractional Riccati initial value problem. This numerical method utilizes the characteristics of Jacobi polynomials to accurately approximate solutions to the nonlinear fractional differential equation. We obtain the requisite Jacobi operational matrices for the discretization of fractional derivatives, therefore converting the initial value problem into a system of algebraic equations. The convergence and precision of the proposed algorithm are meticulously evaluated by error and residual analysis. The theoretical findings demonstrate that the method attains high-order convergence rates, dependent on suitable criteria related to the fractional-order parameters and the solution's smoothness. This study not only improves comprehension of RL circuit dynamics but also offers a solid numerical foundation for addressing intricate fractional differential equations.

Comparison of the combinations of Fourier pseudospectral method, finite volume method, Euler method and fourth order Runge-Kutta method used to solve Burgers equation

The Burgers equation is a mathematical model frequently used in Computational Fluid Dynamics. It is often employed to test and calibrate numerical methods, as it is one of the few nonlinear transport equations with an exact analytical solution. In this paper, numerical solutions are obtained using the Finite Difference Method (FDM) and the Fourier Pseudospectral Method (FPSM) for spatial discretization, combined with the Euler Method and the Fourth-Order Runge-Kutta Method (FRKM) for time discretization. The results are compared with the exact analytical solution in terms of accuracy, convergence rate, and computational cost. The findings indicate that the combination of the FDM and Euler Method achieves excellent computational efficiency when compared to the other approaches. Meanwhile, the combination of FPSM and FRKM demonstrates superior accuracy (achieving round-off errors) and a high order of convergence (spectral convergence order). Thus, combining methods with similar convergence rates and accuracy is the optimal strategy for obtaining efficient numerical solutions of partial differential equations (PDEs).

Numerical analysis of an age-structured model for HIV viral dynamics with latently infected T cells based on collocation methods

In this paper, we consider the numerical threshold for an age-structured HIV model with latently infected T cells. Based on the continuous collocation methods, a semi-discrete scheme is constructed by discretizing the age variable and a numerical basic reproduction number Rh is provided. With the study of higher-order convergence to the real basic reproduction number R0, the relations between Rh and local stability of disease-free are presented. From the viewpoint of full discretization, an equivalent block-Leslie matrix expression is obtained by embedding into a piecewise-discontinuous polynomial space rather than the piecewise-continuous polynomial space. An implicit full-discrete scheme is considered based on a linearly implicit Euler (IMEX) method, of which the computational cost is almost the same as an explicit scheme. It is more important that the dynamical behavior of the age-semi-discretization system is also preserved for any time step whenever Rh is the threshold for the numerical dynamical system of the age-semi-discretization. Finally, numerical applications are shown to HIV models to illustrate our analysis.

Local Convergence Study for an Iterative Scheme with a High Order of Convergence

In this paper, we address a key issue in Numerical Functional Analysis: to perform the local convergence analysis of a fourth order of convergence iterative method in Banach spaces, examining conditions on the operator and its derivatives near the solution to ensure convergence. Moreover, this approach provides a local convergence ball, within which initial estimates lead to guaranteed convergence with details about the radii of domain of convergence and estimates on error bounds. Next, we perform a comparative study of the Computational Efficiency Index (CEI) between the analyzed scheme and some known iterative methods of fourth order of convergence. Our ultimate goal is to use these theoretical findings to address practical problems in engineering and technology.

An Improved Two‐Step Method for Inverse Eigenvalue Problems

ABSTRACTIn this article, we investigate the numerical solutions of the inverse eigenvalue problem (IEP). Motivated by the techniques of Aishima in 2018, we propose here an improved two‐step method for the IEP. Compared with other existing methods, the proposed method has higher convergence order and/or requires less operations. Under the assumption that the relative generalized Jacobian matrices at a solution are nonsingular, the proposed method is proved to be convergent with cubic root‐convergence rate. Numerical examples demonstrate the effectiveness of the improved method.

An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type

Abstract For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $\frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

Structure-preserving finite element methods for computing dynamics of rotating Bose-Einstein condensate

This work is concerned with the construction and analysis of structure-preserving Galerkin methods for computing the dynamics of rotating Bose-Einstein condensate (BEC) based on the Gross-Pitaevskii equation with angular momentum rotation. Due to the presence of the rotation term, constructing finite element methods (FEMs) that preserve both mass and energy remains an unresolved issue, particularly in the context of nonconforming FEMs. Furthermore, in comparison to existing works, we provide a comprehensive convergence analysis, offering a thorough demonstration of the methods’ optimal and high-order convergence properties. Finally, extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical method for rotating BEC, and the quantized vortex lattice’s behavior is scrutinized through a series of numerical tests.

Explicit exponential Runge–Kutta methods for semilinear time-fractional integro-differential equations

In this work, we consider and analyze explicit exponential Runge–Kutta methods for solving semilinear time-fractional integro-differential equation, which involves two nonlocal terms in time. Firstly, the temporal Runge–Kutta discretizations follow the idea of exponential integrators. Subsequently, we utilize the spectral Galerkin method to introduce a fully discrete scheme. Then, we mainly focus on discussing the one-stage and two-stage methods for solving the proposed semilinear problem. Based on special abstract settings, we perform the convergence analysis for the proposed two different stage methods. In this process, we heavily use estimates about the operator family {S̃(t)}, and in combination with Lipschitz continuous condition. Finally, some numerical experiments confirm theoretical results. Meanwhile, applying this scheme to the related linear problem yields high-order convergence, highlighting the advantages of explicit exponential Runge–Kutta methods.

Quadrature based innovative techniques concerning nonlinear equations having unknown multiplicity

Solution of nonlinear equations is one of the most frequently encountered issue in engineering and applied sciences. Most of the intricateed engineering problems are modeled in the frame work of nonlinear equation f(x)=0. The significance of iterative algorithms executed by computers in resolving such functions is of paramount importance and undeniable in contemporary times. If we study the simple roots and the roots having multiplicity greater of any nonlinear equations we come to the point that finding the roots of nonlinear equations having multiplicity greater than one is not trivialvia classical iterative methods. Instability or slow convergence rate is faced by these methods, and also sometimes these methods diverge. In this article, we give some innovative and robust iterative techniques for obtaining the approximate solution of nonlinear equations having multiplicity m>1. Quadrature formulas are implemented to obtain iterative techniques for finding roots of nonlinear equations having unknown multiplicity. The derived methods are the variants of modified Newton method with high order of convergence and better accuracy. The convergence criteria of the new techniques are studied by using Taylor series method. Some examples are tested for the sack of implementations of these techniques. Numerical and graphical comparison shows the performance and efficiency of these new techniques.

Numerical simulation of differential-algebraic equations with embedded global optimization criteria

We are considering differential-algebraic equations with embedded optimization criteria (DAEOs), in which the embedded optimization problem is solved by global optimization. This leads to differential inclusions for cases in which there are multiple global optimizers at the same time. Jump events from one global optimum to another result in nonsmooth DAEs and reduce the order of convergence of the numerical integrator to first-order. Implementation of event detection and location as introduced in this work preserves the higher-order convergence behaviour of the integrator. This allows the computation of discrete tangent and adjoint sensitivities for optimal control problems.

Derivative free processes of high order for nondifferentiable equations in Banach spaces

In this article, we consider two parametric classes of derivative free iterative methods of high order of convergence in Banach spaces. We study local and semilocal convergence results in Banach spaces for both families of iterative processes. We carry out a numerical application to solve a nonlinear and nondifferentiable Fredholm integral equation in the Banach space of continuous functions with the maximum norm.

Numerical study of variable order model arising in chemical processes using operational matrix and collocation method

This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.

Analysis of a line method for reaction-diffusion models of nonlocal type

The considerable interest in the recent literature for nonlocal equations and their applications to a wide range of scientific problems does not appear to be supported by a corresponding advancement in the efforts toward reliable numerical techniques for their solution. The aim of this paper is to provide such an algorithm and above all to prove its high order convergence. The numerical scheme is applied to a diffusion problem in a biological context, arising in population theory. Several simulations are carried out. At first the empirical order of convergence on examples with known solution is assessed. Their results are in agreement with the theoretical findings. Simulations are then extended to cases for which the solution is not known a priori. Also in this case the outcomes support the convergence analysis.

On a Sixth-order Solver for Multiple Roots of Nonlinear Equations

A number of iterative schemes with high convergence order to solve nonlinear equations are presented in the literature. In this paper, a sixth-order multiple-zero finder has been developed and the dynamics of selected iterative schemes with uniparametric polynomial weight function are investigated using M\"{o}bius conjugacy map applied to the form $((z-A)(z-B))^m$. The complex dynamics on the Riemann sphere by analyzing the parameter spaces associated with the free critical points are studied and the numerical experiments are carried out.

A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space

The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in Rm spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects.

High-order genuinely multidimensional finite volume methods via kernel-based WENO

In this paper a family of fully multidimensional kernel-based reconstruction schemes for use in finite volume methods (FVMs) will be presented. These methods are intended for use in shock dominated problems, and stability is achieved through a suitable adaptation of the Adaptive Order Weighted Essentially Non-Oscillatory (WENO-AO) method to the proposed kernel-based reconstruction schemes. There are a number of key difficulties in the design of high-order finite volume schemes which will be discussed and addressed. High (4th and 6th) order convergence will be demonstrated on smooth exact solutions of the ideal MHD equations. The very same scheme will then be applied to extremely stringent astrophysical benchmark problems.

The Boyle–Romberg Trinomial Tree, a Highly Efficient Method for Double Barrier Option Pricing

Oscillations in option price convergence have long been a problematic aspect of tree methods, inhibiting the use of repeated Richardson extrapolation that could otherwise greatly accelerate convergence, a feature integral to some of the most efficient modern methods. These oscillations are typically caused by the fluctuating positions of nodes around the discontinuities in the payoff function or its derivatives. Our paper addresses this crucial gap that typically prohibits the use of lattice methods when high efficiency is needed. Focusing on double barrier options, we develop a trinomial tree in which the positions of the nodes are precisely adjusted to align with these discontinuities throughout the option’s lifespan and across various time steps. This alignment enables the use of repeated extrapolation to achieve high order convergence, including near barriers, a well-known challenge in many tree methods. Maintaining the inherent simplicity and adaptability of tree methods, our approach is easily applicable to other models and option types.

Implicit-explicit time integration for the immersed wave equation

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation.We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.