Nonlinear concentric water waves of moderate amplitude

We consider the outward-propagating nonlinear concentric water waves within the scope of the 2D Boussinesq system. The problem is axisymmetric, and we derive the slow radius versions of the cylindrical Korteweg - de Vries (cKdV) and extended cKdV (ecKdV) models. Numerical runs are initially performed using the full axisymmetric Boussinesq system. At some distance away from the origin, we use the numerical solution of the Boussinesq system as the "initial condition" for the derived cKdV and ecKdV models. We then compare the evolution of the waves as described by both reduced models and the direct numerical simulations of the axisymmetric Boussinesq system. The main conclusion of the paper is that the extended cKdV model provides a much more accurate description of the waves and extends the range of validity of the weakly-nonlinear modelling to the waves of moderate amplitude.Comment: 29 pages, 17 figure

The automatic solution of partial differential equations using a global spectral method

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank $2$, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree $(n_x,n_y)$ is computed in $\mathcal{O}((n_x n_y)^{3/2})$ operations. Partial differential operators of splitting rank $\geq 3$ are solved via a linear system involving a block-banded matrix in $\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3}))$ operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under $60$ seconds. An experimental implementation in the Julia language can currently perform the same solve in $10$ seconds.Comment: 22 page

Components of Nonlinear Oscillation and Optimal Averaging for Stiff PDEs

A novel solver which uses finite wave averaging to mitigate oscillatory stiffness is proposed and analysed. We have found that triad resonances contribute to the oscillatory stiffness of the problem and that they provide a natural way of understanding stability limits and the role averaging has on reducing stiffness. In particular, an explicit formulation of the nonlinearity gives rise to a stiffness regulator function which allows for analysis of the wave averaging. A practical application of such a solver is also presented. As this method provides large timesteps at comparable computational cost but with some additional error when compared to a full solution, it is a natural choice for the coarse solver in a Parareal-style parallel-in-time method

Autoregressive Renaissance in Neural PDE Solvers

Recent developments in the field of neural partial differential equation (PDE) solvers have placed a strong emphasis on neural operators. However, the paper "Message Passing Neural PDE Solver" by Brandstetter et al. published in ICLR 2022 revisits autoregressive models and designs a message passing graph neural network that is comparable with or outperforms both the state-of-the-art Fourier Neural Operator and traditional classical PDE solvers in its generalization capabilities and performance. This blog post delves into the key contributions of this work, exploring the strategies used to address the common problem of instability in autoregressive models and the design choices of the message passing graph neural network architecture.Comment: Presented as a workshop poster at ICLR 202

Applied Mathematics and Computational Physics

As faster and more efficient numerical algorithms become available, the understanding of the physics and the mathematical foundation behind these new methods will play an increasingly important role. This Special Issue provides a platform for researchers from both academia and industry to present their novel computational methods that have engineering and physics applications

NIAC Phase I Final Report: On-Orbit, Collision-Free Mapping of Small Orbital Debris

Sub-centimeter orbital debris is currently undetectable using ground-based radar and optical methods. However, the pits in Space Shuttle windows produced by paint chips (e.g. the 3.8mm diameter pit produced by a 0.2mm paint chip on STS-7) demonstrate that small debris can cause serious damage to spacecraft. Recent analytical, computational and experimental work has shown that charged objects moving quickly through a plasma will cause the formation of solitons in the plasma density. Due to their exposure to the solar wind plasma environment, even the smallest space debris will be charged. Depending on the debris size, charge and velocity, the plasma signature of the solitons may be detected by simple instrumentation on spacecraft. We will describe the amplitude and velocity of solitons that may be produced by mm-cm scale orbital debris in LEO. We will discuss the feasibility of mapping sub-cm orbital debris using a fleet of CubeSats equipped with Langmuir probes. The time and fleet size required to map the debris will also be described. Plasma soliton detection would be the first collision-free method of mapping the small debris population

A multi-domain implementation of the pseudo-spectral method and compact finite difference schemes for solving time-dependent differential equations

Abstract : In this dissertation, we introduce new numerical methods for solving time-dependant differential equations. These methods involve dividing the domain of the problem into multiple sub domains. The nonlinearity of the differential equations is dealt with by using a Gauss-Seidel like relaxation or quasilinearisation technique. To solve the linearized iteration schemes obtained we use either higher order compact finite difference schemes or spectral collocation methods and we call the resulting methods the multi-domain compact finite difference relaxation method (MD-CFDRM), multi-domain compact finite difference quasilinearisation method (MD-CFDQLM) and multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) respectively. We test the applicability of these methods in a wide variety of differential equations. The accuracy is compared against other methods as well as other results from literature. The MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. Chaotic and hyperchaotic systems are characterized by high sensitivity to small perturbation on initial data and rapidly changing solutions. Such rapid variations in the solution pose tremendous problems to a number of numerical approximations. We modify the CFDs to be able to deal with such systems of equations. We also used the MD-CFDQLM to solve the nonlinear evolution partial differential equations, namely, the Fisher’s equation, Burgers- Fisher equation, Burgers-Huxley equation and the coupled Burgers’ equations over a large time domain. The main advantage of this approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method for large time domain. We also studied the generalized Kuramoto-Sivashinsky (GKS) equations. The KS equations exhibit chaotic behaviour under certain conditions. We used the multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) to approximate the numerical solutions for the generalized KS equations.M.Sc. (Pure and Applied Mathematics