scholarly journals A class of one-step time integration schemes for second-order hyperbolic differential equations

2001 ◽  
Vol 33 (4-5) ◽  
pp. 431-443 ◽  
Author(s):  
M.M. Chawla ◽  
M.A. Al-Zanaidi
Keyword(s):  
Differential Equations ◽  
Time Integration ◽  
Second Order ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
One Step ◽  
Hyperbolic Differential Equations

Low-dissipation low-dispersion explicit Taylor-Galerkin schemes from the Runge-Kutta kernels

2018 ◽  
Vol 17 (1-2) ◽  
pp. 88-113
Author(s):  
Mostafa Najafiyazdi ◽  
Luc Mongeau ◽  
Siva Nadarajah
Keyword(s):  
Time Integration ◽  
Second Order ◽  
Order Of Accuracy ◽  
Galerkin Scheme ◽  
Integration Schemes ◽  
Low Dissipation ◽  
Multi Stage ◽  
Time Integration Schemes ◽  
Runge Kutta ◽  
Low Dispersion

A multi-stage approach was adopted to investigate similarities and differences between the explicit Taylor-Galerkin and the explicit Runge-Kutta time integration schemes. It was found that the substitution of some, but not all, of second-order temporal derivatives in a Taylor-Galerkin scheme by additional stages makes it analogous to a Runge-Kutta scheme while preserving its original dissipative property for node-to-node oscillations. The substitution of all second-order temporal derivatives transforms Taylor-Galerkin schemes into Runge-Kutta schemes with zero attenuation at the grid cut-off. The application of this approach to an existing two-stage Taylor-Galerkin scheme yields a low-dissipation low-dispersion Taylor-Galerkin formulation. Two one-dimensional benchmarks were simulated to study the performance of this new scheme. The reverse process yields a general approach for transforming m-stage Runge-Kutta schemes into ( m−1)-stage Taylor-Galerkin schemes while preserving the same order of accuracy. The dissipation and dispersion properties for several new Taylor-Galerkin schemes were compared to those of their corresponding Runge-Kutta form.

scholarly journals Implicit–Explicit Runge–Kutta Methods for Fast–Slow Wave Problems

2013 ◽  
Vol 141 (10) ◽  
pp. 3426-3434 ◽  
Author(s):  
Jeffrey S. Whitaker ◽  
Sajal K. Kar
Keyword(s):  
Slow Wave ◽  
Time Integration ◽  
Forecast Model ◽  
Second Order ◽  
Water Model ◽  
Fast Wave ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
Runge Kutta ◽  
Wave Problems

Abstract Linear multistage (Runge–Kutta) implicit–explicit (IMEX) time integration schemes for the time integration of fast-wave–slow-wave problems for which the fast wave has low amplitude and need not be accurately simulated are investigated. The authors focus on three-stage, second-order schemes and show that a scheme recently proposed by one of them (Kar) is unstable for purely oscillatory problems. The instability is reduced if the averaging inherent in the implicit part of the scheme is decentered, sacrificing second-order accuracy. Two alternative schemes are proposed with better stability properties for purely oscillatory problems. One of these utilizes a 3-cycle Lorenz scheme for the slow-wave terms and a trapezoidal scheme for the fast-wave terms. The other is a combination of two previously proposed schemes, which is stable for purely oscillatory problems for all fast-wave frequencies when the slow-wave frequency is less than a critical value. The alternative schemes are tested using a global spectral shallow-water model and a version of the NCEP operational global forecast model. The accuracy and stability of the alternative schemes are discussed, along with their computational efficiency.

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