Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

Takei, Yuki; Iwata, Yoritaka

doi:10.3390/axioms11010028

Cited by 6 publications

(4 citation statements)

References 27 publications

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“…On the other hand, they have disadvantages such as high computational cost compared to conventional methods. The advent of spectral methods has made it possible to combine high accuracy with low computational cost [1,38,55], they are methods characterized by exponential convergence to the exact solution with increasing grid size.…”

Section: Introductionmentioning

confidence: 99%

Coupling of the immersed boundary and Fourier pseudo-spectral methods applied to solve fluid–structure interaction problems

Nascimento,

Mariano,

da Silveira Neto

et al. 2024

J Braz. Soc. Mech. Sci. Eng.

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The current manuscript addresses fluid–structure interaction (FSI) problems to the analysis of vortex-induced vibration (VIV), employing the methodology designated as IMERSPEC2D. The approach seamlessly integrates the Fourier pseudo-spectral method (FPSM) and the immersed boundary method (IBM) to solve the Navier–Stokes (NS) and continuity equations. Simultaneously, the effects of cylindrical structure, involving 1 and 2 degrees of freedom (d.o.f.) are simulated, by incorporating structural motion equations into NS via the IBM source term and implementing a two-way FSI algorithm. In the paper are presented the comparison of low and high-order temporal integration methods applied to solve the dynamic behavior of the structure and the dimensionless of the cylinder’s structural motion equations, resulting in an increase in computational time step from 1E($$-9$$ - 9 ) to 1E($$-3$$ - 3 ). Validation results to refined mesh simulations include the accurate representation of the lock-in phenomenon to simulations of 1 d.o.f. obtaining the range between reduced velocity 5.456 and 5.568 and the maximal amplitude is 0.352. Notably, the eight-shape pattern of the 2 d.o.f. cylinder displacement is obtained and the formation of C2S wake patterns is achieved, as presented by other authors. In conclusion, the dimensionless approach reduces computational time, while the high-order both of IMERSPEC methodology and time advancement integration to FSI improve the results of the cylinder’s motion and alters distinct vortex shedding patterns in fluid dynamics.

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Section: Introductionmentioning

confidence: 99%

Coupling of the immersed boundary and Fourier pseudo-spectral methods applied to solve fluid–structure interaction problems

Nascimento,

Mariano,

da Silveira Neto

et al. 2024

J Braz. Soc. Mech. Sci. Eng.

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“…Moreover, they have effective error estimators, which can provide an estimate of the local or global error of the numerical solution. These features make IRK methods suitable for optimizing the time step needed to ensure stable and accurate solutions while maintaining the dispersion and dissipation at fixed levels [19][20][21]. Implicit Runge-Kutta (IRK) methods are a cutting-edge group of schemes within the range of advanced time discretization schemes [22].…”

Section: Introductionmentioning

confidence: 99%

Solving a System of One-Dimensional Hyperbolic Delay Differential Equations Using the Method of Lines and Runge-Kutta Methods

Karthick,

Subburayan,

Agarwal

2024

Computation

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In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O(Δt+h¯4), where Δt is the time step and h¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method.

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“…Because of the aforementioned applications, various numerical techniques for solving various application problems, such as time-frequency analysis, signal delay, convection-diffusion equations, nonlinear approximation, and Monte Carlo simulation, arising in fluid dynamics problems have been developed. For instance, predictor-corrector techniques (Su et al [11], Awoyemi and Idowu [12], Iskandarov and Komartsova [13], Ashry et al [14], Asif [15]); Galerkin methods (Guo et al [16]); Haar wavelets methods (Aziz and Khan [17], Shiralashetti et al [18], Saparova et al [19]); Runge-Kutta methods (Takei and Iwata [20], Yakubu et al [21], Zhao and Huang [22]); Newtral network model (Mall and Chakraverty [23]); multigrid technique (Ghaffar et al [24], Ge [25], Gupta et al [26]); finite difference methods (Mulla et al [27]); and finite element methods (Harari and Hughes [28]).…”

Section: Introductionmentioning

confidence: 99%

Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies

Omole,

Adeyefa,

Ayodele

et al. 2023

Axioms

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A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.

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Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

Cited by 6 publications

References 27 publications

Coupling of the immersed boundary and Fourier pseudo-spectral methods applied to solve fluid–structure interaction problems

Coupling of the immersed boundary and Fourier pseudo-spectral methods applied to solve fluid–structure interaction problems

Solving a System of One-Dimensional Hyperbolic Delay Differential Equations Using the Method of Lines and Runge-Kutta Methods

Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies

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