Solving the quintic by iteration in three dimensions

The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the quintic, is that of the symmetric group S_5. Induced by its five-dimensional linear permutation representation is a three-dimensional projective action. A mapping of complex projective 3-space with this S_5 symmetry can provide the requisite symmetry-breaking tool. The article describes some of the S_5 geometry in CP^3 as well as several maps with particularly elegant geometric and dynamical properties. Using a rational map in degree six, it culminates with an explicit algorithm for solving a general quintic. In contrast to the Doyle-McMullen procedure - three 1-dimensional iterations, the present solution employs one 3-dimensional iteration.Comment: 40 pages, 15 figure

Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids

We present a polynomial multigrid method for nodal interior penalty and local discontinuous Galerkin formulations of the Poisson equation on Cartesian grids. For smoothing we propose two classes of overlapping Schwarz methods. The first class comprises element-centered and the second face-centered methods. Within both classes we identify methods that achieve superior convergence rates, prove robust with respect to the mesh spacing and the polynomial order, at least up to ${P=32}$. Consequent structure exploitation yields a computational complexity of $O(PN)$, where $N$ is the number of unknowns. Further we demonstrate the suitability of the face-centered method for element aspect ratios up to 32

Nonuniformly weighted Schwarz smoothers for spectral element multigrid

A hybrid Schwarz/multigrid method for spectral element solvers to the Poisson equation in $\mathbb R^2$ is presented. It extends the additive Schwarz method studied by J. Lottes and P. Fischer (J. Sci. Comput. 24:45--78, 2005) by introducing nonuniform weight distributions based on the smoothed sign function. Using a V-cycle with only one pre-smoothing, the new method attains logarithmic convergence rates in the range from 1.2 to 1.9, which corresponds to residual reductions of almost two orders of magnitude. Compared to the original method, it reduces the iteration count by a factor of 1.5 to 3, leading to runtime savings of about 50 percent. In numerical experiments the method proved robust with respect to the mesh size and polynomial orders up to 32. Used as a preconditioner for the (inexact) CG method it is also suited for anisotropic meshes and easily extended to diffusion problems with variable coefficients.Comment: Multigrid method; Schwarz methods; spectral element method; p-version finite element metho

A numerical method for computing time-periodic solutions in dissipative wave systems

A numerical method is proposed for computing time-periodic and relative time-periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi-Rayleigh quotients, so that the resulting integro-differential equation is for the time-periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton-conjugate-gradient iterations. The proposed method applies to both stable and unstable time-periodic solutions; its numerical accuracy is spectral; it is fast-converging; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto-Sivashinsky equation and the cubic-quintic Ginzburg-Landau equation, whose time-periodic or relative time-periodic solutions with spatially-periodic or spatially-localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in appendices to illustrate the simple implementation of the proposed method.Comment: 24 pages, 5 figure

Dual-Time Smoothed Particle Hydrodynamics for Incompressible Fluid Simulation

In this paper we propose a dual-time stepping scheme for the Smoothed Particle Hydrodynamics (SPH) method. Dual-time stepping has been used in the context of other numerical methods for the simulation of incompressible fluid flows. Here we provide a scheme that combines the entropically damped artificial compressibility (EDAC) along with dual-time stepping. The method is accurate, robust, and demonstrates up to seven times better performance than the standard weakly-compressible formulation. We demonstrate several benchmarks showing the applicability of the scheme. In addition, we provide a completely open source implementation and a reproducible manuscript.Comment: 44 pages, 21 figure

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture

Global well-posedness and scattering for the defocusing H˙12\dot{H}^{\frac{1}{2}}-critical nonlinear Schr\"odinger equation in R2\mathbb{R}^2

In this paper we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schr\"odinger equation in $\mathbb{R}^2$ with general data in the critical space $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$. We show that if a solution remains bounded in $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$ in its maximal interval of existence, then the interval is infinite and the solution scatters

Coherent Cavitation in the Liquid of Light

We study the cubic- (focusing-)quintic (defocusing) nonlinear Schr\"odinger equation in two transverse dimensions. We discuss a family of stationary traveling waves, including rarefaction pulses and vortexantivortex pairs, in a background of critical amplitude. We show that these rarefaction pulses can be generated inside a flattop soliton when a smaller bright soliton collides with it. The fate of the evolution strongly depends on the relative phase of the solitons. Among several possibilities, we find that the dark pulse can reemerge as a bright soliton

Solving the octic by iteration in six dimensions

Extends previous work on a quintic-solving algorithm to equations of the eighth-degree.Comment: 33 pages. arXiv admin note: text overlap with arXiv:math/990305

Numerical Continuation and SPDE Stability for the 2D Cubic-Quintic Allen-Cahn Equation

We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a stochastic $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand, how numerical continuation methods can be carried over to the SPDE setting. First, we compute the deterministic bifurcation diagram for the PDE, i.e. without stochastic forcing. In this case, two locally asymptotically stable steady state solution branches exist upon variation of the linear damping term. Then we consider the Lyapunov operator equation for the locally linearized system around steady states for the SPDE. We discretize the full SPDE using a combination of finite-differences and spectral noise approximation obtaining a finite-dimensional system of stochastic ordinary differential equations (SODEs). The large system of SODEs is used to approximate the Lyapunov operator equation via covariance matrices. The covariance matrices are numerically continued along the two bifurcation branches. We show that we can quantify the stochastic fluctuations along the branches. We also demonstrate scaling laws near branch and fold bifurcation points. Furthermore, we perform computational tests to show that, even with a sub-optimal computational setup, we can quantify the subexponential-timescale fluctuations near the deterministic steady states upon stochastic forcing on a standard desktop computer setup. Hence, the proposed method for numerical continuation of SPDEs has the potential to allow for rapid parametric uncertainty quantification of spatio-temporal stochastic systems.Comment: revised version, 30 pages, 11 figures [movie not included due to arXiv size limitations