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Linearly implicit methods for Ordinary Differential Equations combined with the application of Approximate Matrix Factorization (AMF) provide efficient numerical methods for the solution of large semi-discrete parabolic Partial Differential Equations in several spatial dimensions. Interesting particular subclasses of such linearly implicit methods are the so-called W-methods and the TASE W-methods recently introduced in González-Pinto et al. (Appl Numer Math, 188:129–145, 2023) with the aim of reducing the computational cost of the TASE Runge–Kutta methods in Bassenne et al. (J Comput Phys 424:109847, 2021) and Calvo et al. (J Comp Phys 436:110316, 2021). In this paper, we study the application of the AMF approach in combination with TASE W-methods. While for AMF W-methods the temporal order of consistency is immediately obtained from that of the underlying W-method, this property needs a more thorough analysis for the newly introduced AMF-TASE W-methods. For these latter methods it is described which are the additional order conditions to be fulfilled and it is shown that the parallel structure of the methods is crucial to retain the order of consistency of the underlying TASE W-method. Numerical experiments are presented in three spatial dimensions to assess the consistency result and to show that the proposed schemes are competitive with other well-known good performing AMF W-methods.

We present and evaluate several explicit, large time-stepping algorithms for the Allen-Cahn equation. Our approach incorporates a stabilization technique and uses Taylor series approximations for exponential functions to develop a family of up to third-order parametric Runge–Kutta schemes that maintain fixed-points and maximum principle for any time step τ > 0 . We also introduce a new relaxation technique that eliminates time delay caused by stabilization. To further decrease the stabilization parameter, we utilize an integrating factor with respect to the stiff linear operator and develop a parametric relaxation integrating factor Runge–Kutta (pRIFRK) framework. Compared to existing maximum-principle-preserving (MPP) schemes, the proposed parametric relaxation approaches are free from limiters, cut-off post-processing, exponential decay, or time delay. Linear stability analysis determines that the parametric approaches are A -stable when appropriate parameters are used. In addition, we provide error estimates in the l ∞ -norm with the help of the MPP property. We demonstrate the high-order temporal accuracy, maximum-principle-preservation, energy stability, and delay-free properties of the proposed schemes through a set of experiments on 1D, 2D, and 3D problems.

This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit (pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on sinc matrix functions. The novelty contained here is that of using a suitable rational approximation formula for the sinc matrix function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we discuss the application of the whole strategy to a finite element discretization of the wave equation.

In this paper, we propose a method for the numerical calculation of the highest Lyapunov exponent for linear ordinary differential systems.

A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge–Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found for non-uniformly refined meshes and spatially varying wave speeds. To demonstrate the applicability of the stability polynomials we construct 2 N -storage many-stage Runge–Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. These methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.

Runge–Kutta embedded pairs of high algebraic order are frequently utilized when strict tolerances are required. When creating such pairings of orders nine and eight for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients’ magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders 9 and 8 with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems.

The stabilization approach has been known to permit large time-step sizes while maintaining stability. However, it may “slow down the convergence rate” or cause “delayed convergence” if the time-step rescaling is not well resolved. By considering a fourth-order-in-space viscous Cahn–Hilliard (VCH) equation, we propose a class of up to the fourth-order single-step methods that are able to capture the correct physical behaviors with high-order accuracy and without time delay. By reformulating the VCH as a system consisting of a second-order diffusion term and a nonlinear term involving the operator ( I - ν Δ ) - 1 , we first develop a general approach to estimate the maximum bound for the VCH equation equipped with either the Ginzburg–Landau or Flory–Huggins potential. Then, by taking advantage of new recursive approximations and adopting a time-step-dependent stabilization, we propose a class of stabilization Runge–Kutta methods that preserve the maximum principle for any time-step size without harming the convergence. Finally, we transform the stabilization method into a parametric Runge–Kutta formulation, estimate the rescaled time-step, and remove the time delay by means of a relaxation technique. When the stabilization parameter is chosen suitably, the proposed parametric relaxation integrators are rigorously proven to be mass-conserving, maximum-principle-preserving, and the convergence in the l ∞ -norm is estimated with p th-order accuracy under mild regularity assumption. Numerical experiments on multi-dimensional benchmark problems are carried out to demonstrate the stability, accuracy, and structure-preserving properties of the proposed schemes.

The matrix-valued Allen-Cahn (MAC) equation was first introduced as a model problem of finding the stationary points of an energy for orthogonal matrix-valued functions and has attracted much attention in recent years. It is well known that the MAC equation satisfies the maximum bound principle (MBP) with respect to either the matrix 2-norm or the Frobenius norm, which plays a key role in understanding the physical meaning and the wellposedness of the model. To preserve this property, we extend the explicit integrating factor Runge-Kutta (IFRK) method to the MAC equation. Moreover, we construct a new three-stage third-order and a new four-stage fourth-order IFRK schemes based on the classical Runge-Kutta schemes. Under a reasonable time-step constraint, we prove the MBP preservation of the IFRK method with respect to the matrix 2-norm, based on which, we further establish their optimal error estimates in the matrix 2-norm. Although numerical results indicate that the IFRK method preserves the MBP with respect to the Frobenius norm, a detailed analysis shows that it is hard to prove this preservation by using the same approach for the case of 2-norm. Several numerical experiments are carried out to test the convergence of the IFRK schemes and to verify the MBP preservation with respect to the matrix 2-norm and the Frobenius norm, respectively. Energy stability is also observed, which clearly indicates the orthogonality of the stationary solution of the MAC equation. In addition, we simulate the coarsening dynamics to verify the motion law of the interface for different initial conditions.

In this paper, we study the long-time stability-preserving properties of the two-step backward differentiation formula (BDF2) fully discrete scheme for the fractional complex Ginzburg-Landau equation. More precisely, we consider the BDF2 time discretization together with a general spatial spectral discretization and show that the numerical scheme can unconditionally preserve the long-time stability in the L 2 , H 1 , H 1 + α and H 2 norms with the aid of the discrete uniform Gronwall lemma. As a special case, we obtain the long-time stability-preserving properties of the fully discrete BDF2 spectral method for the standard complex Ginzburg-Landau equation for the first time. Numerical examples are presented to support our theoretical results.