Communications in Applied Mathematics and Computational Science最新文献

A novel multifluid plasma model is developed, based on the finite-volume method, with a fourth-order accurate algorithm and solution-adaptive mesh refinement in space and time. Previously, the order of accuracy of the multifluid plasma model was verified and the solutions to common plasma test cases were demonstrated. In the present work, the multifluid plasma model is applied to solve complex test cases involving discontinuities and shocks. Discussion is thus focused on methods of numerical stabilization and strategies with adaptive mesh refinement. The results show improvements to solution stability and as much as an order of magnitude improvement in compute time due to adaptive mesh refinement.

Physics-informed neural networks (PINNs) can be used not only to predict the solutions of nonlinear partial differential equations, but also to discover the dynamic characteristics and phase transitions of rogue waves in nonlinear systems. Based on improved PINNs, we predict bright-dark one-soliton, two-soliton, two-soliton molecule and rogue wave solutions in a coupled AB system. We find that using only a small number of dynamic evolutionary rogue wave solutions as training data, we can find the phase transition boundary that can distinguish bright and dark rogue waves, and realize the mutual prediction between different rogue wave structures. The results show that the improved algorithm has high prediction accuracy, which provides a promising general technique for discovering and predicting new rogue structures in other parametric coupled systems.

We combine the recent relaxation approach with multiderivative Runge–Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.

We present and analyze a parallel implementation of a parallel-in-time collocation method based on $\alpha$-circulant preconditioned Richardson iterations. While many papers explore this family of single-level, time-parallel “all-at-once” integrators from various perspectives, performance results of actual parallel runs are still scarce. This leaves a critical gap, because the efficiency and applicability of any parallel method heavily rely on the actual parallel performance, with only limited guidance from theoretical considerations. Further, challenges like selecting good parameters, finding suitable communication strategies, and performing a fair comparison to sequential time-stepping methods can be easily missed. In this paper, we first extend the original idea of these fixed point iterative approaches based on $\alpha$-circulant preconditioners to high-order collocation methods, adding yet another level of parallelization in time “across the method”. We derive an adaptive strategy to select a new $\alpha$-circulant preconditioner for each iteration during runtime for balancing convergence rates, round-off errors, and inexactness of inner system solves for the individual time-steps. After addressing these more theoretical challenges, we present an open-source space- and time-parallel implementation and evaluate its performance for two different test problems.

We present a study of the standard plasma physics test, Landau damping, using the particle-in-cell (PIC) algorithm. The Landau damping phenomenon consists of the damping of small oscillations in plasmas without collisions. In the PIC method, a hybrid discretization is constructed with a grid of finitely supported basis functions to represent the electric, magnetic and/or gravitational fields, and a distribution of delta functions to represent the particle field. Approximations to the dispersion relation are found to be inadequate in accurately calculating values for the electric field frequency and damping rate when parameters of the physical system, such as the plasma frequency or thermal velocity, are varied. We present a full derivation and numerical solution for the dispersion relation, and verify the PETSC-PIC numerical solutions to the Vlasov–Poisson system for a large range of wavenumbers and charge densities.

We investigate the achievable efficiency of exact solvers for the Riemann problem for two systems of first-order hyperbolic PDEs: the shallow water equations and the Euler equations of compressible gas dynamics. Many approximate solvers have been developed for these systems; exact solution algorithms have received less attention because the computation of the exact solution typically requires an iterative solution of algebraic equations, which can be expensive or unreliable. We investigate a range of iterative algorithms and initial guesses. In addition to existing algorithms, we propose simple new algorithms that are guaranteed to converge and to remain in the range of physically admissible values at all iterations. We apply the existing and new iterative schemes to an ensemble of test Riemann problems. For the shallow water equations, we find that Newton’s method with a simple modification converges quickly and reliably. For the Euler equations we obtain similar results; however, when the required precision is high, a combination of Ostrowski and Newton iterations converges faster. These solvers are slower than standard approximate solvers like Roe and HLLE, but come within a factor of two in speed. We also provide a preliminary comparison of the accuracy of a finite volume discretization using an exact solver versus standard approximate solvers.

Gaussian process regression is a widely applied method for function approximation and uncertainty quantification. The technique has recently gained popularity in the machine learning community due to its robustness and interpretability. The mathematical methods we discuss in this paper are an extension of the Gaussian process framework. We are proposing advanced kernel designs that only allow for functions with certain desirable characteristics to be elements of the reproducing kernel Hilbert space (RKHS) that underlies all kernel methods and serves as the sample space for Gaussian process regression. These desirable characteristics reflect the underlying physics; two obvious examples are symmetry and periodicity constraints. In addition, we want to draw attention to nonstationary kernel designs that can be defined in the same framework to yield flexible multitask Gaussian processes. We will show the impact of advanced kernel designs on Gaussian processes using several synthetic and two scientific data sets. The results show that informing a Gaussian process of domain knowledge, combined with additional flexibility and communicated through advanced kernel designs, has a significant impact on the accuracy and relevance of the function approximation.