5,423 research outputs found
More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence
Time integration of Fourier pseudo-spectral DNS is usually performed using
the classical fourth-order accurate Runge--Kutta method, or other methods of
second or third order, with a fixed step size. We investigate the use of
higher-order Runge-Kutta pairs and automatic step size control based on local
error estimation. We find that the fifth-order accurate Runge--Kutta pair of
Bogacki \& Shampine gives much greater accuracy at a significantly reduced
computational cost. Specifically, we demonstrate speedups of 2x-10x for the
same accuracy. Numerical tests (including the Taylor-Green vortex,
Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the
reliability and efficiency of the method. We also show that adaptive time
stepping provides a significant computational advantage for some problems (like
the development of a Rayleigh-Taylor instability) without compromising
accuracy
RKN-type parallel block PC methods with Lagrange-type predictors
AbstractThis paper describes the construction of block predictor-corrector methods based on Runge-Kutta-Nyström correctors. Our approach is to apply the predictor-corrector method not only at step points, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-order predictor formula using Lagrange interpolation. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictor formulas based on last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta-Nyström corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor-corrector methods to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature
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