An asymptotic parallel-in-time method for highly oscillatory PDEs (1303.6615v2)

Abstract: We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which, alone, can be inefficient for equations that exhibit rapid temporal oscillations). In particular, we use an asymptotic numerical method for computing, in serial, a solution with low accuracy, and a more expensive fine solver for iteratively refining the solutions in parallel. We present examples on the rotating shallow water equations that demonstrate that significant parallel speedup and high accuracy are achievable.

• The paper introduces a hybrid algorithm that integrates the parareal method with asymptotic HMM techniques to address oscillatory dynamics.
• It demonstrates significant speedup, with roughly 100x improvement over standard methods and a tenfold advantage over conventional parareal algorithms.
• The method mitigates strict small time-step constraints in nonlinear oscillatory PDEs, promising enhanced efficiency in large-scale geophysical simulations.

An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

The paper presents a new time-stepping algorithm specifically tailored for nonlinear partial differential equations (PDEs) with highly oscillatory temporal scales. The primary contribution is the integration of the parareal method, a parallel-in-time computational scheme, with techniques derived from the Heterogeneous Multiscale Method (HMM). This hybrid approach effectively leverages the slow asymptotic structure of the equations to mitigate the constraints on time step size typically imposed by fast oscillations.

Model and Methodology

The paper considers PDEs characterized by equations of the form:

$$\frac{\partial\mathbf{u}}{\partial t} + \frac{1}{\epsilon}\mathcal{L}\mathbf{u} = \mathcal{N}(\mathbf{u}) + \mathcal{D}\mathbf{u}, \quad \mathbf{u}(0) = \mathbf{u}_0,$$

where $\mathcal{L}$ is a linear operator inducing oscillations with pure imaginary eigenvalues, $\mathcal{N}$ is a nonlinear polynomial, $\mathcal{D}$ represents dissipation, and $\epsilon$ is a small parameter highlighting the fast oscillatory nature. The challenge addressed by this research arises from the necessity for extremely small time steps when employing standard numerical integrators, due to the rapid oscillations driven by $\epsilon^{-1}\mathcal{L}$.

The authors innovatively extend the parareal framework by substituting the numerical coarse solver with an asymptotic approximation of the full equations, thereby circumventing the Nyquist constraint dictated by the fastest oscillations. The coarse approximation is computed using a 'locally slow' solution, grounded in the asymptotic properties of the governing equations, allowing the use of significantly larger coarse time steps compared to existing methods.

Analysis and Results

Error bounds cited from literature ensure convergence of the proposed method, and a complexity analysis reveals that parallel speedup increases with greater scale separation. Specifically, the paper demonstrates an estimated parallel speedup of approximately 100 times when applied to the standard rotating shallow water equations in one dimension, a canonical test for new algorithms in the geophysical domain. This performance is notably superior to that of exponential integrators like ETDRK4 and traditional splitting methods, where the latter are constrained to use much smaller time steps to achieve comparable accuracy.

The numerical experiments also quantify a significant advantage for the asymptotic parareal method, achieving a factor of 10 greater speedup than the standard parareal algorithm across a range of physically relevant parameters. Notably, when $\epsilon = 10^{-2}$, the results underscore the method's efficacy in enhancing efficiency without sacrificing accuracy, and its robustness across a spectrum of temporal scale separations.

Implications and Future Work

The proposed algorithm holds potential to markedly enhance computational efficiency in large-scale geophysical simulations, where the limits of spatial parallelization are being approached. Extending these capabilities to multidimensional and more complex geophysical flows opens avenues for further research. Additionally, integrating these methods with adaptive mesh refinement or hybrid CPU-GPU computing frameworks could further optimize performance in high-resolution simulation environments.

Looking forward, the theoretical framework and numerical strategies derived herein could be extended to a broader class of multiscale problems beyond geophysical fluid dynamics, reinforcing their utility in sophisticated multiscale modeling endeavors in computational physics and engineering. The continuation of this research should focus on expanding the operational range and efficiency of the method, especially in handling non-periodic boundary conditions and variable-coefficient PDEs that are prevalent in real-world applications.