Local Truncation Error Research Articles

Truncation Error of the Network Simulation Method: Chaotic Dynamical Systems in Mechanical Engineering

This article focuses on the study of local truncation errors (LTEs) in the Network Simulation Method (NSM), specifically when using the trapezoidal method and Gear’s methods. The NSM, which represents differential equations through electrical circuit elements, offers advantages in solving nonlinear dynamic systems such as the van der Pol equation. The analysis compares the performance of these numerical methods in terms of their stability and error minimization, with particular emphasis on LTE. By leveraging circuit-based techniques prior to numerical application, the NSM improves convergence. This study evaluates the impact of step size on LTE and highlights the trade-offs between accuracy and computational cost when using the trapezoidal and Gear methods.

About One Multistep Multiderivative Method of Predictor-Corrector Type Constructed for Solving İnitial-Value Problem for ODE of Second Order

Considering the wide application of the initial-value problem for Ordinary Differential Equations second-order with a special structure, here for solving this problem constructed the special Multistep Multiderivative Methods. Many scientists studied this problem , but the most distinguishing is the Ştörmer. To solve this problem here is proposed to use the Multistep Second derivative Method with a special structure. This method has been generalized by many authors, which is called as the linear Multistep Multiderivative Methods with the constant coefficients. Many authors shave shown that the Multistep Second derivative Method can be applied to solve the initial-value problem for ODEs of the first order. Euler himself using his famous method discovered that, in his method when moving from one point to another local truncation errors add up, the results of which reach a very large value. To solve this problem, he suggested using more accurate methods. For this aim, Euler proposed calculating the next term in the Taylor series of the solutions of the investigated problem. Developing this idea and papulation of the Multistep Multiderivative Methods here to solve the named problem it is suggested to use MultistepThriedderivative Methods, taking into account that methods of this type are more accurate. For the demonstration above, receiving results here have constructed some concrete methods. Also by using some of Dahlquist’s and Ibrahimov’s results for Multistep Methods with the maximum order of accuracy were compared. Proven that the MultistepThriedderivative Methods are more accurate than the others. By using model problems have illustrated some results received here.

High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

Optimal error bounds of the time-splitting sine-pseudospectral method for the biharmonic nonlinear Schrödinger equation

We propose a time-splitting sine-pseudospectral (TSSP) method for the biharmonic nonlinear Schrödinger equation (BNLS) and establish its optimal error bounds. In the proposed TSSP method, we adopt the sine-pseudospectral method for spatial discretization and the second-order Strang splitting for temporal discretization. The proposed TSSP method is explicit and structure-preserving, such as time symmetric, mass conservation and maintaining the dispersion relation of the original BNLS in the discretized level. Under the assumption that the solution of the one dimensional BNLS belongs to Hm with m≥9, we prove error bounds at O(τ2+hm) and O(τ2+hm−1) in L2 norm and H1 norm respectively, for the proposed TSSP method, with τ time step and h mesh size. For general dimensional cases with d=1,2,3, the error bounds are at O(τ2+hm) and O(τ2+hm−2) in L2 and H2 norm under the assumption that the exact solution is in Hm with m≥10. The proof is based on the bound of the Lie-commutator for the local truncation error, discrete Gronwall inequality, energy method and the H1- or H2-bound of the numerical solution which implies the L∞-bound of the numerical solution. Finally, extensive numerical results are reported to confirm our optimal error bounds and to demonstrate rich phenomena of the solutions including rapidly dispersion in space of high frequency waves and soliton collisions.

An explicit fourth‐order hybrid‐variable method for Euler equations with a residual‐consistent viscosity

AbstractIn this article, we present a formally fourth‐order accurate hybrid‐variable (HV) method for the Euler equations in the context of method of lines. The HV method seeks numerical approximations to both cell averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are supraconvergent, that is, the order of the method is higher than that of the local truncation error. Taking advantage of the supraconvergence, the method is built on a third‐order discrete differential operator, which approximates the first spatial derivative at each grid point using only the information in the two neighboring cells. Analyses of stability, accuracy, and pointwise convergence are conducted in the one‐dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual‐consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems.

Adaptive conservative time integration for unsteady compressible flow

Unsteady flow computations typically rely on time integration schemes which employ the same step size everywhere. However, in cases with strong variations of wave speed and/or spatial resolution, local stability criteria and truncation errors would allow for much larger time steps in parts of the domain. To exploit this potential of improving the computational efficiency, adaptive time-stepping methods have been developed. Recently, an adaptive conservative time integration (ACTI) scheme for explicit finite volume methods was devised. Opposed to previous methods, ACTI guarantees exact conservation and periodic synchronization. Both are achieved by splitting a global time step into 2L (L≥0 is a cell dependent level) sub-steps, whereas the order of cell updates is critical. It has been demonstrated for flows in fractured porous media and for the Euler equations that ACTI can reduce the computational cost dramatically while preserving second order accuracy in space and time. In this paper, the ACTI scheme is extended to the compressible Navier-Stokes-Fourier system, and special attention is required for the spatio-temporal discretization of the convective and viscous fluxes. Numerical studies of unsteady flows involving shock boundary layer interaction demonstrate that the same accuracy is achieved with the new compressible ACTI flow solver as with classical time integration, but at much lower cost. Moreover, since the method is explicit, it has the potential for efficient computations on multiple CPUs and GPUs.

Triangular finite differences using bivariate Lagrange polynomials with applications to elliptic equations

This paper proposes finite-difference schemes based on triangular stencils to approximate partial derivatives using bivariate Lagrange polynomials. We use first-order partial derivative approximations on triangles to introduce a novel hexagonal scheme for the second-order partial derivative on any rotated parallelogram grid. Numerical analysis of the local truncation errors shows that first-order partial derivative approximations depend strongly on the triangle vertices getting at least a first-order method. On the other hand, we prove that the proposed hexagonal scheme is always second-order accurate. Simulations performed at different triangular configurations reveal that numerical errors agree with our theoretical results. Results demonstrate that the proposed method is second-order accurate for the Poisson and Helmholtz equation. Furthermore, this paper shows that the hexagonal scheme with equilateral triangles results in a fourth-order accurate method to the Laplace equation. Finally, we study two-dimensional elliptic differential equations on different triangular grids and domains.

Krylov-based adaptive-rank implicit time integrators for stiff problems with application to nonlinear Fokker-Planck kinetic models

We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of solutions with significantly reduced computational costs. We further introduce an efficient mechanism for residual evaluation and an adaptive rank-seeking strategy that optimizes low-rank settings based on a comparison between the residual size and the local truncation errors of the time-stepping discretization. We demonstrate our approach with the challenging Lenard-Bernstein Fokker-Planck (LBFP) nonlinear equation, which describes collisional processes in a fully ionized plasma. The preservation of the equilibrium state is achieved through the Chang-Cooper discretization, and strict conservation of mass, momentum and energy via a Locally Macroscopic Conservative (LoMaC) procedure. The development of implicit adaptive-rank integrators, demonstrated here up to third-order temporal accuracy via diagonally implicit Runge-Kutta schemes, showcases superior performance in terms of accuracy, computational efficiency, equilibrium preservation, and conservation of macroscopic moments. This study offers a starting point for developing scalable, efficient, and accurate methods for high-dimensional time-dependent problems.

The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

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(τ 2 + h 2) 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(τ 4 + h 4) in L ∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.

From low-rank retractions to dynamical low-rank approximation and back

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston–Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

A New Fractional Representation of the Higher Order Taylor Scheme

In this work, we suggest a new numerical scheme called the fractional higher order Taylor method (FHOTM) to solve fractional differential equations (FDEs). Using the generalized Taylor’s theorem is the fundamental concept of this approach. Then, the local truncation error generated by the suggested FHOTM is estimated by proving suitable theoretical results. At last, several numerical applications are given to demonstrate the applicability of the suggested approach in relation to their exact solutions.

Hexagonal finite differences for the two-dimensional variable coefficient Poisson equation

For many years, finite differences in hexagonal grids have been developed to solve elliptic problems such as the Poisson and Helmholtz equations. However, these schemes are limited to constant coefficients, which reduces their usefulness in many applications. The main challenge is accurately approximating the diffusive term. This paper presents examples of both successful and unsuccessful attempts to obtain accurate finite differences based on a hexagonal stencil with equilateral triangles to approximate two-dimensional Poisson equations. Local truncation error analysis reveals that a second-order scheme can be achieved if the derivative of the diffusive coefficient is included. Finally, we provide numerical examples to verify the accuracy of the proposed methods.

A novel stochastic ten non-polynomial cubic splines method for heat equations with noise term

In this paper, a new numerical method for solving a class of stochastic partial differential equations is presented. The proposed method is based on a non-polynomial cubic spline algorithm with an O(h4) local truncation error. The new approach has the advantage of simultaneously including ten non-polynomial cubic spline schemes. The proposed approach is also tested on three stochastic heat equations. The current results are compared to solutions obtained using the B-spline wavelet approximation method. The provided solutions demonstrate the proposed method's accuracy and applicability.

Finite element method for a generalized constant delay diffusion equation

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results.

Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation

Abstract The two-dimensional conformal time-fractional generalized q q -deformed sinh-Gordon equation has been used to model a variety of physical systems, including soliton propagation in asymmetric media, nonlinear waves in optical fibers, quantum field theory, and condensed matter physics. The equation is able to capture the complex dynamics of these systems and has been shown to be a powerful tool for studying them. This article discusses the two-dimensional conformal time-fractional generalized q q -deformed sinh-Gordon equation both analytically and numerically using Kudryashov’s approach and the finite difference method. In addition, the stability analysis and local truncation error of the equation are discussed. A number of illustrations are also included to show the various solitons propagation patterns. The proposed equation has opened up new possibilities for modeling asymmetric physical systems.

Numerical coupling of aerosol emissions, dry removal, and turbulent mixing in the E3SM Atmosphere Model version 1 (EAMv1) – Part 2: A semi-discrete error analysis framework for assessing coupling schemes

Abstract. Part 1 (Wan et al., 2024) of this study discusses the motivation and empirical evaluation of a revision to the aerosol-related numerical process coupling in the atmosphere component of the Energy Exascale Earth System Model version 1 (EAMv1) to address the previously reported issue of strong sensitivity of the simulated dust aerosol lifetime and dry removal rate to the model's vertical resolution. This paper complements that empirical justification of the revised scheme with a mathematical justification leveraging a semi-discrete analysis framework for assessing the splitting error of process coupling methods. The framework distinguishes the error due to numerical splitting from the error due to the time integration method(s) used for each individual process. Such a distinction results in a framework that provides an intuitive understanding of the causes of the splitting error. The application of this framework to the dust life cycle in EAMv1 confirms (i) that the original EAMv1 scheme artificially strengthens the effect of dry removal processes and (ii) that the revised splitting reduces that artificial strengthening. While the error analysis framework is presented in the context of the dust life cycle in EAMv1, the framework can be broadly leveraged to evaluate process coupling schemes, both in other physical problems and for any number of processes. This framework will be particularly powerful when the various process implementations support a variety of time integration approaches. Whereas traditional local truncation error approaches require separate consideration of each combination of time integration methods, this framework enables evaluation of coupling schemes independent of particular time integration approaches for each process while still allowing for the incorporation of these specific time integration errors if so desired. The framework also explains how the splitting error terms result from (i) the integration of individual processes in isolation from other processes and (ii) the choices of input state and time step size for the isolated integration of processes. Such a perspective has the potential for the rapid development of alternative coupling approaches that utilize knowledge both about the desired accuracy and about the computational costs of individual processes.

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

A one-step three-parameter optimized hybrid block method and second derivative hybrid block method with optimized points were proposed to solve first-order ordinary differential equations. The techniques of interpolation and collocation were adopted for the derivation of the methods using a three-parameter approximation. The hybrid points were obtained by optimizing the local truncation error of the method. The schemes obtained were reformulated to reduce the number of occurrences of the source term. The hybrid points were used in the derivation of the second derivative hybrid block method. The discrete schemeswere produced as a by-product of the continuous scheme and used to simultaneously solve initial value problems (IVPs) in block mode. The resulting schemes are self-starting, do not require the creation of individual predictors, and are consistent, zero-stable, and convergent. The accuracy and efficiency of the methods were ascertained using several numerical test problems. The numerical results were favourably compared to some techniques from the cited literature.

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

Multiquadric based RBF-HFD approximation formulas and convergence properties

RBF-FD formulas have received special attention due to their superior accuracy, well-conditioning, and ease of implementation in solving partial differential equations. To further improve the accuracy of RBF-FD formulas, without increasing the stencil size RBF-HFD formulas are proposed in the literature. In this article, we obtain leading/higher order analytical expressions for weights and local truncation errors associated with various such formulas using the multiquadric radial function. We also study the convergence properties of these formulas. Symbolic computation in Mathematica is employed for analytically solving linear systems for the unknown weights in the RBF-HFD formulas. Symmetry properties of central difference approximations help reduce the number of unknown weights and thereby the size of the linear system. We compute the analytical expressions of weights and local truncation errors for first and second-derivative approximation formulas till tenth order; and for two-dimensional Laplacian operator approximation formulas till sixth order. The obtained formulas are free from ill-conditioning. It is observed that the RBF-HFD formula weights converge to corresponding order classical compact FD scheme weights when the shape parameter (ϵ) tends to zero. The local truncation error has global minimum for an optimal value of the shape parameter. Further, the optimal shape parameter is independent of nodal distance but depends on choice of test function and its derivative values at a reference node.

Numerical integration of stiff problems using a new time-efficient hybrid block solver based on collocation and interpolation techniques

In this study, an optimal L-stable time-efficient hybrid block method with a relative measure of stability is developed for solving stiff systems in ordinary differential equations. The derivation resorts to interpolation and collocation techniques over a single step with two intermediate points, resulting in an efficient one-step method. The optimization of the two off-grid points is achieved by means of the local truncation error (LTE) of the main formula. The theoretical analysis shows that the method is consistent, zero-stable, seventh-order convergent for the main formula, and L-stable. The highly stiff systems solved with the proposed and other algorithms (even of higher-order than the proposed one) proved the efficiency of the former in the context of several types of errors, precision factors, and computational time.