arXiv subjects

Marcel Oliver

Publications and source records attributed to Marcel Oliver.

17 recordsLinked to original sources

arXiv2024

A new construction of modified equations for variational integrators

The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified equations with increasingly high frequencies which increase the regularity requirements on the analysis. In this paper, we consider the next order modified equations for the implicit midpoint rule applied to the semilinear wave equation to give a proof-of-concept of a new construction which works directly with the variational principle. We show that a carefully chosen change of coordinates yields a modified system which inherits its analytical properties from the original wave equation. Our method systematically exploits additional degrees of freedom by modifying the symplectic structure and the Hamiltonian together.

math.NA↗

arXiv2023

Deterministic and stochastic surrogate models for a slowly driven fast oscillator

It has long been known that the excitation of fast motion in certain two-scale dynamical systems is linked to the singularity structure in complex time of the slow variables. We demonstrate, in the context of a fast harmonic oscillator forced by one component of the Lorenz 1963 model, that this principle can be used to construct time-discrete surrogate models by numerically extracting approximate locations and residues of complex poles via Adaptive Antoulas-Andersen (AAA) rational interpolation and feeding this information into the known "connection formula" to compute the resulting fast amplitude. Despite small but nonnegligible local errors, the surrogate model maintains excellent accuracy over very long times. In addition, we observe that the long-time behavior of fast energy offers a continuous-time analog of Gottwald and Melbourne's 2004 "0-1 test for chaos" - the asymptotic growth rate of the energy in the oscillator can discern whether or not the forcing function is chaotic.

nlin.AO↗

arXiv2022

Quasi-convergence of an implementation of optimal balance by backward-forward nudging

Optimal balance is a non-asymptotic numerical method to compute a point on the slow manifold for certain two-scale dynamical systems. It works by solving a modified version of the system as a boundary value problem in time, where the nonlinear terms are adiabatically ramped up from zero to the fully nonlinear dynamics. A dedicated boundary value solver, however, is often not directly available. The most natural alternative is a nudging solver, where the problem is repeatedly solved forward and backward in time and the respective boundary conditions are restored whenever one of the temporal end points is visited. In this paper, we show quasi-convergence of this scheme in the sense that the termination residual of the nudging iteration is as small as the asymptotic error of the method itself, i.e., under appropriate assumptions exponentially small. This confirms that optimal balance in its nudging formulation is an effective algorithm. Further, it shows that the boundary value problem formulation of optimal balance is well posed up at most a residual error as small as the asymptotic error of the method itself. The key step in our proof is a careful two-component Gronwall inequality.

math.DS↗

arXiv2021

Conditional uniqueness of solutions to the Keller-Rubinow model for Liesegang rings in the fast reaction limit

We study the question of uniqueness of weak solution to the fast reaction limit of the Keller and Rubinow model for Liesegang rings as introduced by Hilhorst et al. (J. Stat. Phys. 135, 2009, pp. 107-132). The model is characterized by a discontinuous reaction term which can be seen as an instance of spatially distributed non-ideal relay hysteresis. In general, uniqueness of solutions for such models is conditional on certain transversality conditions. For the model studied here, we give an explicit description of the precipitation boundary which gives rise to two scenarios for non-uniqueness, which we term "spontaneous precipitation" and "entanglement". Spontaneous precipitation can be easily dismissed by an additional, physically reasonable criterion in the concept of weak solution. The second scenario is one where the precipitation boundaries of two distinct solutions cannot be ordered in any neighborhood of some point on their common precipitation boundary. We show that for a finite, possibly short interval of time, solutions are unique. Beyond this point, unique continuation is subject to a spatial or temporal transversality condition. The temporal transversality condition takes the same form that would be expected for a simple multicomponent semilinear ODE with discontinuous reaction terms.

math.AP↗

arXiv2020

Variational balance models for the three-dimensional Euler-Boussinesq equations with full Coriolis force

We derive a semi-geostrophic variational balance model for the three-dimensional Euler--Boussinesq equations on the non-traditional $f$-plane under the rigid lid approximation. The model is obtained by a small Rossby number expansion in the Hamilton principle, with no other approximations made. We allow for a fully non-hydrostatic flow and do not neglect the horizontal components of the Coriolis parameter, i.e., we do not make the so-called "traditional approximation". The resulting balance models have the same structure as the "$L_1$ balance model" for the primitive equations: a kinematic balance relation, the prognostic equation for the three-dimensional tracer field, and an additional prognostic equation for a scalar field over the two-dimensional horizontal domain which is linked to the undetermined constant of integration in the thermal wind relation. The balance relation is elliptic under the assumption of stable stratification and sufficiently small fluctuations in all prognostic fields.

math.AP↗

arXiv2020

Breakdown of Liesegang precipitation bands in a simplified fast reaction limit of the Keller-Rubinow model

We study solutions to the integral equation \[ ω(x) = Γ- x^2 \int_{0}^1 K(θ) \, H(ω(xθ)) \, \mathrm d θ\] where $Γ>0$, $K$ is a weakly degenerate kernel satisfying, among other properties, $K(θ) \sim k \, (1-θ)^σ$ as $θ\to 1$ for constants $k>0$ and $σ\in (0, \log_2 3 -1)$, $H$ denotes the Heaviside function, and $x \in [0,\infty)$. This equation arises from a reaction-diffusion equation describing Liesegang precipitation band patterns under certain simplifying assumptions. We argue that the integral equation is an analytically tractable paradigm for the clustering of precipitation rings observed in the full model. This problem is nontrivial as the right hand side fails a Lipschitz condition so that classical contraction mapping arguments do not apply. Our results are the following. Solutions to the integral equation, which initially feature a sequence of relatively open intervals on which $ω$ is positive ("rings") or negative ("gaps") break down beyond a finite interval $[0,x^*]$ in one of two possible ways. Either the sequence of rings accumulates at $x^*$ ("non-degenerate breakdown") or the solution cannot be continued past one of its zeroes at all ("degenerate breakdown"). Moreover, we show that degenerate breakdown is possible within the class of kernels considered. Finally, we prove existence of generalized solutions which extend the integral equation past the point of breakdown.

math.CA↗

arXiv2020

High-order uniformly accurate time integrators for semilinear wave equations of Klein-Gordon type in the non-relativistic limit

We introduce a family of high-order time semi-discretizations for semilinear wave equations of Klein--Gordon type with arbitrary smooth nonlinerities that are uniformly accurate in the non-relativistic limit where the speed of light goes to infinity. Our schemes do not require pre-computations that are specific to the nonlinearity and have no restrictions in step size. Instead, they rely upon a general oscillatory quadrature rule developed in a previous paper (Mohamad and Oliver, arXiv:1909.04616).

math.NA↗

arXiv2019

Long-time asymptotics of solutions to the Keller-Rubinow model for Liesegang rings in the fast reaction limit

We consider the Keller--Rubinow model for Liesegang rings in one spatial dimension in the fast reaction limit as introduced by Hilhorst, van der Hout, Mimura, and Ohnishi in 2007. Numerical evidence suggests that solutions to this model converge, independent of the initial concentration, to a universal profile for large times in parabolic similarity coordinates. For the concentration function, the notion of convergence appears to be similar to attraction to a stable equilibrium point in phase space. The reaction term, however, is discontinuous so that it can only convergence in a much weaker, averaged sense. This also means that most of the traditional analytical tools for studying the long-time behavior fail on this problem. In this paper, we identify the candidate limit profile as the solution of a certain one-dimensional boundary value problem which can be solved explicitly. We distinguish two nontrivial regimes. In the first, the transitional regime, precipitation is restricted to a bounded region in space. We prove that the concentration converges to a single asymptotic profile. In the second, the supercritical regime, we show that the concentration converges to one of a one-parameter family of asymptotic profiles, selected by a solvability condition for the one-dimensional boundary value problem. Here, our convergence result is only conditional: we prove that if convergence happens, either pointwise for the concentration or in an averaged sense for the precipitation function, then the other field converges likewise; convergence in concentration is uniform, and the asymptotic profile is indeed the profile selected by the solvability condition. A careful numerical study suggests that the actual behavior of the equation is indeed the one suggested by the theorem.

math.AP↗

arXiv2019

Numerical integration of functions of a rapidly rotating phase

We present an algorithm for the efficient numerical evaluation of integrals of the form \[ I(ω) = \int_0^1 F( x,\mathrm e^{\mathrm i ωx}; ω) \, \mathrm d x \] for sufficiently smooth but otherwise arbitrary $F$ and $ω\gg 1$. The method is entirely "black-box", i.e., does not require the explicit computation of moment integrals or other pre-computations involving $F$. Its performance is uniform in the frequency $ω$. We prove that the method converges exponentially with respect to its order when $F$ is analytic and give a numerical demonstration of its error characteristics.

math.NA↗

arXiv2018

Geodesic motion on the groups of diffeomorphisms with $H^1$ metric as geometric generalised Lagrangian mean theory

Generalized Lagrangian mean theories are used to analyze the interactions between mean flows and fluctuations, where the decomposition is based on a Lagrangian description of the flow. A systematic geometric framework was recently developed by Gilbert and Vanneste (J. Fluid Mech., 2018) who cast the decomposition in terms of intrinsic operations on the group of volume preserving diffeomorphism or on the full diffeomorphism group. In this setting, the mean of an ensemble of maps can be defined as the Riemannian center of mass on either of these groups. We apply this decomposition in the context of Lagrangian averaging where equations of motion for the mean flow arise via a variational principle from a mean Lagrangian, obtained from the kinetic energy Lagrangian of ideal fluid flow via a small amplitude expansion for the fluctuations. We show that the Euler-$α$ equations arise as Lagrangian averaged Euler equations when using the $L^2$-geodesic mean on the volume preserving diffeomorphism group of a manifold without boundaries, imposing a `Taylor hypothesis', which states that first order fluctuations are transported as a vector field by the mean flow, and assuming that fluctuations are statistically isotropic. Similarly, the EPDiff equations arise as the Lagrangian averaged Burgers' equations using the same argument on the full diffeomorphism group. These results generalize an earlier observation by Oliver (Proc. R. Soc. A, 2017) to manifolds in geometrically fully intrinsic terms.

math-ph↗

arXiv2018

Geometric Lagrangian averaged Euler-Boussinesq and primitive equations

In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler-Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler-Poincaré equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler-Boussinesq-$α$ and primitive equations-$α$ models, however feature a different regularizing second order operator.

physics.flu-dyn↗

arXiv2017

Optimal balance via adiabatic invariance of approximate slow manifolds

We analyze the method of optimal balance which was introduced by Viudez and Dritschel (J. Fluid Mech. 521, 2004, pp. 343-352) to provide balanced initializations for two-dimensional and three-dimensional geophysical flows, here in the simpler context of a finite dimensional Hamiltonian two-scale system with strong gyroscopic forces. It is well known that when the potential is analytic, such systems have an approximate slow manifold that is defined up to terms that are exponentially small with respect to the scale separation parameter. The method of optimal balance relies on the observation that the approximate slow manifold remains an adiabatic invariant under slow deformations of the nonlinear interactions. The method is formulated as a boundary value problem for a homotopic deformation of the system from a linear regime, where the slow-fast splitting is known exactly, to the full nonlinear regime. We show that, providing the ramp function which defines the homotopy is of Gevrey class 2 and satisfies vanishing conditions to all orders at the temporal end points, the solution of the optimal balance boundary value problem yields a point on the approximate slow manifold that is exponentially close to the approximation to the slow manifold via exponential asymptotics, albeit with a smaller power of the small parameter in the exponent. In general, the order of accuracy of optimal balance is limited by the order of vanishing derivatives of the ramp function at the temporal end points. We also give a numerical demonstration of the efficacy of optimal balance, showing the dependence of accuracy on the ramp time and the ramp function.

math.DS↗

arXiv2017

Comparison of variational balance models for the rotating shallow water equations

We present an extensive numerical comparison of a family of balance models appropriate to the semi-geostrophic limit of the rotating shallow water equations, and derived by variational asymptotics in Oliver (2006) for small Rossby numbers ${\mathrm{Ro}}$. This family of generalized large-scale semi-geostrophic (GLSG) models contains the $L_1$-model introduced by Salmon (1983) as a special case. We use these models to produce balanced initial states for the full shallow water equations. We then numerically investigate how well these models capture the dynamics of an initially balanced shallow water flow. It is shown that, whereas the $L_1$-member of the GLSG family is able to reproduce the balanced dynamics of the full shallow water equations on time scales of ${\mathcal{O}}(1/{\mathrm{Ro}})$ very well, all other members develop significant unphysical high wavenumber contributions in the ageostrophic vorticity which spoil the dynamics.

physics.flu-dyn↗

arXiv2012

Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations

We prove that a class of A-stable symplectic Runge--Kutta time semidiscretizations (including the Gauss--Legendre methods) applied to a class of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. As a consequence, such time-semidiscretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is $O(h^p)$-close to the original energy where $p$ is the order of the method and $h$ the time step-size. Examples of such systems are the semilinear wave equation or the nonlinear Schr\"odinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace where the operators occurring in the evolution equation are bounded and by coupling the number of excited modes as well as the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form $O(\exp(-c/h^{1/(1+q)}))$ with $c>0$ and $q \geq 0$; for the semilinear wave equation, $q=1$, and for the nonlinear Schr\"odinger equation, $q=2$. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.

math.NA↗

arXiv2010

A-stable Runge-Kutta methods for semilinear evolution equations

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, $A$-stable Runge--Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation.

math.NA↗

arXiv2010

Stability under Galerkin truncation of A-stable Runge--Kutta discretizations in time

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow by an implicit, A-stable Runge--Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semiflow and its time-discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. Then we estimate the Galerkin truncation error for the semiflow of the evolution equation, its Runge--Kutta discretization, and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation.

math.NA↗

arXiv1999

The vortex blob method as a second-grade non-Newtonian fluid

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$\alpha$ equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in ${\mathcal M}({\mathbb R}^2)$ such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods.

math.AP↗