Courant-Friedrichs-Lewy Research Articles

A Novel Approach Integrating the Method of Characteristics with Large Time-Step Scheme (MOC-LTS) for Efficient Transient Flow Simulation in Liquid Pipelines

Water hammer incidents pose a significant risk to the safety and stability of pipeline operations. Therefore, rapid and precise transient flow simulation is essential for efficiently developing scientific water hammer control strategies. Nevertheless, the prevailing transient flow simulation methods for liquid pipelines predominantly employ explicit schemes to solve transient flow control equations, necessitating adherence to the stability criterion of Courant-Friedrichs-Lewy (CFL) ≤ 1. This results in limited time step sizes, which in turn constrains computational efficiency, particularly when simulating hydraulic behavior in large-scale pipeline networks. In this paper, a novel approach integrating the method of characteristics (MOC) with a large time-step scheme (LTS) is proposed to enable rapid and accurate simulation of transient flow in liquid pipelines. The proposed approach discretizes the computational domain into contiguous control volumes, ategorizing them as either boundary or internal control volumes depending on whether they are affected by boundary conditions within a large time step. For internal control volumes, an LTS scheme based on the first-order Godunov format is employed to improve computational efficiency. For boundary control volumes, MOC is applied iteratively to accurately capture boundary dynamics until the simulated time matches that of the internal control volumes. Quantitative analysis is conducted to verify the performance of the proposed approach. The results confirm that the proposed approach outperforms the MOC in computational efficiency while maintaining minimal accuracy loss.

Three-dimensional numerical simulation and experimental validation of airborne ground-penetrating radar based on CNCS-FDTD method under undulating terrain conditions

Traditional Finite Difference Time Domain (FDTD) approaches face challenges with increased computational demands and errors as terrain complexity and flight altitude rising. This study introduces the Crank-Nicolson-Cycle-Sweep-FDTD (CNCS-FDTD) method, enhancing airborne ground penetrating radar (GPR) simulations over undulating terrains. CNCS-FDTD, as an unconditionally stable implicit algorithm, overcomes these by allowing larger time steps without the constraints of the Courant-Friedrichs-Lewy (CFL) condition. Our research aims to assess how CNCS-FDTD can improve computational efficiency and accuracy in modeling airborne GPR responses across varied terrains. Initial simulations using a three-dimensional graben model indicate that CNCS-FDTD can maintain calculation stability with significantly larger time steps, reducing computational time by 42%. To further verify the reliability of the numerical simulation results, the paper also presents experimental tests of the undulating terrain model. By comparing the numerical and physical simulation results of models under different flight altitudes and terrain conditions, the accuracy of the simulation results is validated.

Finite element modelling of linear rolling contact problems

The present work is devoted to the finite element modelling of linear hyperbolic rolling contact problems. The main equations encountered in rolling contact mechanics are reviewed in the first part of the paper, with particular emphasis on applications from automotive and vehicle engineering. In contrast to the common hyperbolic systems found in the literature, such equations include integral and boundary terms, as well as time-varying transport velocities, that require special treatment. In this context, existence and uniqueness properties are discussed within the theoretical framework offered by the semigroup theory. The second part of the paper is then dedicated to recovering approximated solutions to the considered problems, by combining discontinuous Galerkin finite element methods (DGMs) with explicit Runge–Kutta (RK) schemes of the first and second order for time discretisation. Under opportune assumptions on the smoothness of the sought solutions, and owing to certain generalised Courant–Friedrichs–Lewy (CFL) conditions, quasi-optimal error bounds are derived for the complete discrete schemes. The proposed algorithms are then tested on simple scalar equations in one space dimension. Numerical experiments seem to suggest the theoretical error estimates to be sharp.

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess both scale-invariant and evolution-invariant properties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate for nonlinear DG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure is nonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques. Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method.

On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation

AbstractWe consider a model initial‐ and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second‐order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal‐order error estimates in the and the norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant‐Friedrichs‐Lewy (CFL) condition between the space mesh size and the time step sizes.

A 3-D Global FDTD Courant-limit Model of the Earth for Long-time-span and High-altitude Applications

A new global 3-D finite-difference time-domain (FDTD) model is introduced to simulate electromagnetic wave propagation around the Earth, including the lithosphere, oceans, atmosphere, and ionosphere regions. This model has several advantages over existing global models, which include grids that follow lines of latitude and longitude and geodesic grids comprised of hexagons and pentagons. The advantages of the new model include: (1) it may be run at the Courant-Friedrichs-Lewy (CFL) time step (as a result, it is termed the Courant-limit model); (2) subgrids may be added to specific regions of the model as needed in a straight-forward manner; and (3) the grid cells do not become infinitely larger as the grid is extended higher in altitude. As a result, this model is a better candidate than the others for investigating electromagnetic phenomena over long time spans of interest and for investigating atmosphere-ionosphere-magnetosphere coupling. The new model is first described and then validated by comparing results for extremely low frequency (ELF) propagation attenuation with corresponding analytical and measurement results reported in the literature.

A hybrid one‐step leapfrog ADI‐FDTD and subgridding approach based on a heterogeneous computing platform

AbstractThe finite‐difference time‐domain (FDTD) method often demands numerous grids for fine structure analysis, leading to high computational costs. To address this challenge, a subgridding scheme emerges as an attractive solution. However, the explicit nature of the FDTD subgrid method, constrained by Courant–Friedrichs–Lewy (CFL) conditions, can lead to inefficiencies with complex structures. In response, the leapfrog alternating‐direction implicit (ADI)‐FDTD method, exhibiting unconditional stability, has been developed to overcome CFL limitations. This paper presents a hybrid approach merging the one‐step leapfrog ADI‐FDTD method with a subgridding scheme using heterogeneous computing. By optimizing hardware usage and leveraging different device performances, this approach maximizes throughput in heterogeneous systems. Central processing units and graphics processing units handle workloads to minimize system latency. Simulation results on heterogeneous platforms demonstrate superior efficiency over single processor setups while maintaining simulation accuracy, especially in analyzing electromagnetic scattering from multiscale structures.

Application of spatial filtering high order FDTD method in the multi-physics field coupled equations

The traditional finite-difference time-domain method (FDTD(2, 2)) is widely employed to solve the coupled equations of quantum-electromagnetic multi-physics. However, the time step of the traditional explicit FDTD(2, 2) method is limited by the Courant-Friedrichs-Lewy (CFL) condition. Besides, the FDTD(2, 2) method only achieves the second-order accuracy, which can result in significant error accumulation. To overcome this problem, we propose the SF-HO-FDTD(2, q) method with scalable time stability conditions for solving the multi-physics field coupled equations (q denotes q-th order spatial difference scheme). This approach integrates spatial filtering (SF) and higher-order FDTD (HO-FDTD) methods. Moreover, the proposed method requires no further derivation of the iterative equations of the traditional HO-FDTD(2, q) method. It only needs to introduce the SF operation in each numerical iteration process to filter out the spatial high-frequency components arising from the use of time steps that do not satisfy the CFL stability condition. This condition is applied to ensure the stability of the numerical method. Therefore, the proposed method has a high compatibility with the traditional HO-FDTD(2, q) method. Then, the numerical dispersion analysis of the SF-HO-FDTD(2, q) method is theoretically studied. Finally, numerical examples are carried out to confirm the accuracy and efficiency of the strategy suggested in this study.

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

A leap-frog nodal discontinuous Galerkin method for Maxwell polynomial chaos Debye model

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method.

Nonlinear diffusion‐limited two‐dimensional colliding‐plume simulations with very high‐order numerical approximations

AbstractAtmospheric numerical models play a crucial role in operational weather forecasting, as well as in improving our understanding of atmospheric dynamics via research studies. Maximising their accuracy is of paramount importance. Use of advective flux schemes greater than O7 in atmospheric models is largely undocumented, with no studies considering O3–O17 fluxes with formal accuracy‐preserving high‐order interpolation, pressure gradient/divergence, and subgrid‐scale (SGS) turbulent fluxes. Higher‐order numerical approximations can reduce truncation, amplitude and phase errors, and potentially improve model accuracy and effective resolution. Here, simulations are presented using very‐high‐order O3–O17 fluxes, with/without high‐order O2–O18 Lagrangian interpolations, pressure gradient/divergence approximations, and SGS turbulent fluxes for a two‐dimensional, highly‐viscous (Re ∼ 100) diffusion‐limited, nonlinear colliding‐plumes problem using 25–200 m spatial resolutions. The highest‐order flux schemes coupled with higher‐order interpolations, pressure gradient/divergence and SGS flux approximations produced the best solutions, with higher‐order fluxes and interpolations being most important. Overall solution convergence of order about O1–O2 with mode‐split (fast sound/slow advective waves) O3 Runge–Kutta temporal schemes was negatively impacted by order at most O1 temporal convergence with SGS fluxes, divergence damping, and especially spatial filters, compared to about order O3 convergence with these inactivated. While very‐high‐order schemes were shown to improve solution accuracy, few cost‐effective higher‐order highly‐viscous test problem solutions (higher order vs finer resolution) were found using theoretical floating‐point operations (FPO) with Courant‐Friedrichs‐Lewy (CFL)‐limited or constant stable Courant number‐based time steps. However, employing central processing unit (CPU) time, rather than FPOs, demonstrated there was reduced computational burden using higher‐order approximations. We conclude that O9–O17 flux schemes with or without high‐order (≥O4) interpolations, pressure gradient/divergence approximations, and SGS fluxes can improve atmospheric model solution accuracy, without prohibitive computation costs, compared to O3–O7 flux with O2 interpolations, pressure gradient/divergence approximations, and SGS fluxes.

Application of high-order SF-SFDTD scheme to solving a time-dependent SchrFDTD scheme , Wu

The finite-difference time-domain 2, 2 (FDTD (2, 2)) method with second-order numerical accuracy in time and space has been extensively employed in the field of quantum mechanics to solve the Schrödinger equation. Nevertheless, the presence of the Courant-Friedrichs-Lewy (CFL) condition imposes limitations on the grid size in the computational space, thereby constraining the admissible range of time steps. Accordingly, the efficiency of the FDTD(2, 2) method significantly decreases. In addition, the second-order numerical accuracy of the FDTD(2, 2) method both in time domain and in space domain often results in significant error accumulation during calculations, thereby undermining the fidelity of the simulation results. To surmount the constraints imposed by the CFL stability conditions and enhance the accuracy of computations, a novel approach termed SF-SFDTD(3, 4) method has been proposed, with 3 and 4 referring to the accuracy in space and time, respectively. This method combines spatial filtering (SF) with the high-order symplectic finite-difference time-domain (SFDTD) method. Its primary objective is to solve the time-dependent Schrödinger equation while ensuring time stability and scalability. The SF-SFDTD(3, 4) method obviates the need for further deriving the iterative formula employed in the conventional SFDTD(3, 4) method. Therefore, the method under consideration exhibits a remarkable degree of compatibility with its traditional counterpart. It is merely necessary to include a spatial filtering operation during each numerical iteration to eliminate spatial high-frequency components arising from the utilization of time step sizes that fail to satisfy the CFL stability condition, thereby ensuring the stability of the numerical scheme. Moreover, when the time step value satisfies the CFL stability condition, the amplitude of the high-frequency component approaches zero, thereby exerting a minimal influence on the accuracy of the computational results. The adoption of time steps that do not meet the CFL stability conditions leads to an amplification in the amplitude of the high-frequency component. However, this finding solely affects the stability of the computational results, and the elimination of these unstable high-frequency components scarcely affect the accuracy of the computational results. The SF-SFDTD(3, 4) retains the simplicity and efficacy inherent in the traditional SFDTD(3, 4) methods, while enhancing computational efficiency. Additionally, the numerical stability and dispersion error of the SF-SFDTD(3, 4) method are analyzed theoretically . Finally, the validity and efficacy of the proposed method are corroborated through numerical illustrations.

A discontinuous Galerkin finite element model for compound flood simulations

Many tropical cyclones, e.g., Hurricane Harvey (2017), have lead to significant rainfall and resulting runoff. When the runoff interacts with storm surge, the resulting floods can be greatly amplified and lead to effects that cannot be correctly modeled by simple superposition of its distinctive sources. In an effort to develop accurate numerical simulations of runoff, surge, and compounding floods, we develop a local discontinuous Galerkin method for modified shallow water equations. In this modification, nonzero sources to the continuity equation are included to incorporate rainfall into the model using parametric rainfall models from literature as well as hindcast data. The discontinuous Galerkin spatial discretization is accompanied with a strong stability preserving explicit Runge Kutta time integrator. Hence, temporal stability is ensured through the Courant–Friedrichs–Lewy (CFL) condition and we exploit the embarrassingly parallel nature of the developed method using MPI parallelization.We demonstrate the capabilities of the developed method though a sequence of physically relevant numerical tests, including small scale test cases based on laboratory measurements and large scale experiments with Hurricane Harvey in the Gulf of Mexico. The results highlight the conservation properties and robustness of the developed method and show the potential of compound flood modeling using our approach.

A coupled level-set and tangent of hyperbola interface capturing (THINC) scheme with a single-step time integration for incompressible flows

In this paper, we present a concise and efficient interface capturing scheme on polyhedral unstructured mesh with its numerical implementation for incompressible fluid simulation with surface tension. The so-called coupled Level-Set and Tangent of Hyperbola INterface Capturing scheme with a Consistent Single-step time integration (THINC/CSLS) updates both the volume-of-fluid (VOF) and level-set function simultaneously from a THINC reconstruction function. The proposed scheme has the potential to integrate the positive aspects of mass conservation from the VOF method and the accurate geometric representation from the level-set method. Other innovative and noteworthy aspects of the present scheme are listed as (1) an adaptive THINC reconstruction strategy is applied to ensure the boundness of the VOF function for various Courant–Friedrichs–Lewy (CFL) numbers and the high quality of the level-set function for surface tension; (2) a single-step update procedure for both VOF and level-set function is implemented for better algorithmic consistency and efficiency; (3) a well-adapted numerical model is developed for incompressible simulation with free surface for both structured and unstructured grids. A comprehensive numerical procedure is introduced and validated by various well-received benchmark tests. Convincing evidence suggests that the present scheme is both accurate and efficient compared with other most advanced methods.

Optimized Traditional and Leapfrog CDI-FDTD Schemes With Reduced Dispersion and Numerical Properties Analysis

In order to improve the numerical dispersion property of the finite-difference time-domain (FDTD) method further, two optimized low-dispersion schemes are herein proposed, which are based on the complying-divergence implicit (CDI) FDTD method and its one-step leapfrog method. These proposed approaches effectively reduce the numerical dispersion errors by intervening with anisotropic parameters while maintaining unconditional stability. The numerical stability and dispersion properties are analyzed in detail to illustrate their effectiveness. The factors, which impact the numerical dispersion including propagation angles, Courant–Friedrichs–Lewy (CFL) number, nonuniform multiscale, mesh resolution, and frequency are carefully examined. In addition, by using the dual-/triple-band multiple-input multiple-output (MIMO) antennas as numerical examples, the practicality of the optimization schemes is comprehensively validated by comparing the calculation time, memory resource, and relative error with those from the traditional and leapfrog CDI-FDTD methods. It is worth celebrating the superiority of the proposed schemes not only in terms of numerical dispersion, but also in calculation efficiency, solution accuracy, and floating-point operands.

A stable staggered-grid finite-difference scheme for acoustic modeling beyond conventional stability limit

Staggered-grid finite-difference (SGFD) schemes have been widely used in acoustic wave modeling for geophysical problems. Many improved methods are proposed to enhance the accuracy of numerical modeling. However, these methods are inevitably limited by the maximum Courant-Friedrichs-Lewy (CFL) numbers, making them unstable when modeling with large time sampling intervals or small grid spacings. To solve this problem, we extend a stable SGFD scheme by controlling SGFD dispersion relations and maximizing the maximum CFL numbers. First, to improve modeling stability, we minimize the error between the FD dispersion relation and the exact relation in the given wave-number region, and make the FD dispersion approach a given function outside the given wave-number area, thus breaking the conventional limits of the maximum CFL number. Second, to obtain high modeling accuracy, we use the SGFD scheme based on the Remez algorithm to compute the FD coefficients. In addition, the hybrid absorbing boundary condition is adopted to suppress boundary reflections and we find a suitable weighting coefficient for the proposed scheme. Theoretical derivation and numerical modeling demonstrate that the proposed scheme can maintain high accuracy in the modeling process and the value of the maximum CFL number of the proposed scheme can exceed that of the conventional SGFD scheme when adopting a small maximum effective wavenumber, indicating that the proposed scheme improves stability during the modeling.

Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors

Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.

A semi-implicit semi-Lagrangian method for simulating immersed boundary motion under high inertia and elasticity

In this paper, we present an efficient and stable fractional-step 2D immersed boundary (IB) method for solving the interaction problems between bulk fluid and elastic interface, in particular, when the fluid inertia and the interfacial elasticity are the significant factors affecting its dynamics. In myriads of real-world applications, the effects of high inertia and elasticity are dominant. So the complex fluid dynamics under such harsh conditions is an important topic in computational physics and is inherently challenging due to high computational complexity. In turn, it requires to solve the governing equations of elastic interfacial motion in an implicit manner so that more stable simulations can be performed by relaxing the Courant-Friedrichs-Lewy (CFL) condition. The contributions of the proposed approach are three folds. First, an iteration-free semi-Lagrangian method is employed in Navier-Stokes (NS) equations. Second, the elastic force acting along the interface is treated semi-implicitly in IB formulations. Both approaches improve the numerical stability associated with the high fluidic inertia and interfacial elasticity. Finally, to solve the resulting linear system, our novel idea is to transform the original 3-by-3 block matrix system into a reduced 2-by-2 block matrix system using a discrete projection operator in a staggered grid, and then explicitly represent the exact solution via the Schur complement of the Helmholtz operator. Owing to this feature, we refer to this proposed approach as reduced immersed boundary method (rIBM). We show that the two systems are equivalent in theory, whereas the conventional immersed boundary projection method (IBPM) modifies the discrete momentum equation in the original system. A series of numerical tests is conducted to confirm the stability of the rIBM using relatively larger time-step sizes, specifically with Reynolds number and inverse capillary number equal to or larger than approximately 1000. By estimating the computational time, the numerical efficiency of the proposed method is further verified in comparison with the conventional IBPM and the Crank-Nicolson scheme-based IB method. In conclusion, the proposed approach not only improves the numerical stability, but also increases the computational speed, suitable for solving more realistic problems.

Auto-tuning numerical method for acoustic wave simulation using analytical solution.

Numerical wavefield simulation such as commercial simulation software enables an optimal design of an ultrasound computed tomography (USCT) system for clinical purpose of before prototyping. Such simulator, not developed for optimal design though, can provide rapid implementation for acoustic wave propagation but may lead to unexpected errors during the establishment of numerical model. Here, we propose an auto-tuning numerical method (ATNM), aiming to optimize physical parameters (e.g. grid size, Courant-Friedrichs-Lewy (CFL) number, perfectly matched layer (PML) absorption coefficient, etc) such that the enumerated wavefield computed on those converges to the corresponding analytical solution derived from acoustic scattering theory. We use genetic algorithm (GA) to automatically calibrate numerical wavefield. Our preliminary test is to investigate the best design of PML absorption coefficient for USCT to minimize mean relative error (MRE) between the k-Wave simulation and the analytic model and show its efficacy. The experimental results verify our hypothesis that this calibrated numerical simulator on a simple physical domain is generalizable to any other domains.

An optimal spatial-filtering method derived from eigenvalue perturbation for extending the Courant-Friedrichs-Lewy stability limit

Explicit time-marching schemes are widely used in numerical simulations. However, their maximum time step is constrained by the Courant-Friedrichs-Lewy (CFL) stability limit. Two methods have been developed to extend this limit: (1) the eigenvalue perturbation method, which exhibits high numerical accuracy but incurs unaffordable memory demand and computational costs, even for middle-scale models, and (2) the spatial-filtering method, which can be implemented easily but results in significant numerical errors under large time steps. However, the intrinsic relationship between these two methods remains unknown. We reveal the intrinsic relationship between these two methods by considering the eigenvalue perturbation method for a homogeneous model. Using this relationship, we derive an analytical spatial filter for the homogeneous model and develop an optimal spatial filter for heterogeneous models. Compared to the classical spatial-filtering method, which removes all high-wavenumber components, our method retains all high-wavenumber components that contribute to wave propagation while eliminating other high-wavenumber components that cause instability. For the same numerical accuracy, the maximum time step allowed by the proposed method is approximately twice that of the classical spatial-filtering method. Compared to the eigenvalue perturbation method, our method can be used in large-scale models without additional memory consumption and computational cost. This can significantly accelerate the pseudospectral method with significantly larger time steps beyond the CFL stability limit, which is particularly promising for large-scale models with a fine grid.