Alexander Bihlo

Alexander Bihlo, Xavier Coiteux-Roy, Pavel Winternitz

The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the Eulerian form. We also propose invariant schemes that preserve the momentum. Numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes. We find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving Galilean transformations which is demonstrated numerically as well.

Alexander Bihlo, Roman O. Popovych

Invariant discretization schemes are derived for the one- and two-dimensional shallow-water equations with periodic boundary conditions. While originally designed for constructing invariant finite difference schemes, we extend the usage of difference invariants to allow constructing of invariant finite volume methods as well. It is found that the classical invariant schemes converge to the Lagrangian formulation of the shallow-water equations. These schemes require to redistribute the grid points according to the physical fluid velocity, i.e., the mesh cannot remain fixed in the course of the numerical integration. Invariant Eulerian discretization schemes are proposed for the shallow-water equations in computational coordinates. Instead of using the fluid velocity as the grid velocity, an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level. The numerical conservation of energy, mass and momentum is evaluated for both the invariant and non-invariant schemes.

Rüdiger Brecht, Werner Bauer, Alexander Bihlo et al.

We develop a variational integrator for the shallow-water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler-Poincaré reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler-Poincaré equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and the excellent conservational properties of the discrete variational integrator.

Alexander Bihlo, Francis Valiquette

In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames. The advantages of both techniques are discussed and illustrated with the Schwarzian differential equation, the Korteweg-de Vries equation and Burgers' equation. Numerical simulations are presented and innovative techniques for obtaining better invariant numerical schemes are introduced. New research directions and open problems are indicated at the end of these notes.

Alexander Bihlo, Scott MacLachlan

We formulate a general criterion for the exact preservation of the "lake at rest" solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. The main idea is a careful mimetic design for the spatial derivative operators in the momentum flux equation that is paired with a compatible averaging rule for the water column height arising in the bottom topography source term. We prove consistency of the mimetic difference operators analytically and demonstrate the well-balanced property numerically using finite difference and RBF-FD schemes in the one- and two-dimensional cases.

Andy T. S. Wan, Alexander Bihlo, Jean-Christophe Nave

We show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order system of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and non-autonomous dynamical systems with conserved quantities of arbitrary forms, such as time-dependent conserved quantities. Sufficient conditions to construct conservative schemes of arbitrary order are derived using the multiplier method. General formulas for first-order conservative schemes are constructed using divided difference calculus. New conservative schemes are found for various dynamical systems such as Euler's equation of rigid body rotation, Lotka-Volterra systems, the planar restricted three-body problem and the damped harmonic oscillator.

Alexander Bihlo, Jean-Christophe Nave

In the recent paper by Bernardini et al. [J. Comput. Phys. 232 (2013), 1-6] the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied. In a simplified investigation carried out using the viscous Burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of Galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow. Here we further discuss this problem and relate it to known results on invariant discretization schemes. Non-invariant and invariant finite difference discretizations of Burgers equation are proposed and compared with the discretization using the remedy proposed by Bernardini et al..

Alexander Bihlo

A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a way of associating invariant functions to non-invariant functions. An invariant meshless approximation of a nonlinear diffusion equation is constructed. Comparative numerical tests with a non-invariant meshless scheme are presented. These tests yield that invariant meshless schemes can lead to substantially improved numerical solutions compared to numerical solutions generated by non-invariant meshless schemes.

Alexander Bihlo, Francis Valiquette

Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our constructions can be extended to (1+1)-dimensional evolutionary partial differential equations, using Burgers' equation as an example. Numerical simulations verify that the symmetry-preserving finite element schemes constructed converge at the expected rate and that these schemes can yield better results than their non-invariant finite element counterparts.

Fabrizio Donzelli, Alexander Bihlo, Mauricio Kischinhevsky et al.

In this paper, we report the advantages of using a stochastic algorithm in the context of mineral exploration based on gravity measurements. This approach has the advantage over deterministic methods in that it allows one to find the solution of the Poisson equation in specified, isolated points without the need of meshing the computational domain and solving the Poisson equation over the entire domain. Moreover, the stochastic approach is embarrassingly parallelizable and therefore suitable for an implementation on multi-core compute clusters with or without GPUs. Benchmark tests are carried out that show that the stochastic approach can yield accurate results for both the gravitational potential and the gravitational acceleration and could hence provide an alternative to existing deterministic methods used in mineral exploration.

Alexander Bihlo, Colin G. Farquharson, Ronald D. Haynes et al.

Stochastic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell's equations as used in the magnetotelluric method. The stochastic form of the exact solution of Maxwell's equations is evaluated using Monte-Carlo methods taking into consideration that the domain may be divided into neighboring sub-domains. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell's equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.

Alexander Bihlo

We use a conditional deep convolutional generative adversarial network to predict the geopotential height of the 500 hPa pressure level, the two-meter temperature and the total precipitation for the next 24 hours over Europe. The proposed models are trained on 4 years of ERA5 reanalysis data from 2015-2018 with the goal to predict the associated meteorological fields in 2019. The forecasts show a good qualitative and quantitative agreement with the true reanalysis data for the geopotential height and two-meter temperature, while failing for total precipitation, thus indicating that weather forecasts based on data alone may be possible for specific meteorological parameters. We further use Monte-Carlo dropout to develop an ensemble weather prediction system based purely on deep learning strategies, which is computationally cheap and further improves the skill of the forecasting model, by allowing to quantify the uncertainty in the current weather forecast as learned by the model.

Alexander Bihlo

We propose the use of a stochastic variational frame prediction deep neural network with a learned prior distribution trained on two-dimensional rain radar reflectivity maps for precipitation nowcasting with lead times of up to 2 1/2 hours. We present a comparison to a standard convolutional LSTM network and assess the evolution of the structural similarity index for both methods. Case studies are presented that illustrate that the novel methodology can yield meaningful forecasts without excessive blur for the time horizons of interest.

Andy T. S. Wan, Alexander Bihlo, Jean-Christophe Nave

We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on discretizing conservation law multipliers and their associated density and flux functions. We show that the proposed discretization is consistent for any order of accuracy when the discrete multiplier has a multiplicative inverse. Moreover, we show that by construction, discrete densities can be exactly conserved. In particular, the multiplier method does not require the system to possess a Hamiltonian or variational structure. Examples, including dissipative problems, are given to illustrate the method. In the case when the inverse of the discrete multiplier becomes singular, consistency of the method is also established for scalar ODEs provided the discrete multiplier and density are zero-compatible.

Alexander Bihlo, Ronald D. Haynes, Emily J. Walsh

The efficient generation of meshes is an important component in the numerical solution of problems in physics and engineering. Of interest are situations where global mesh quality and a tight coupling to the solution of the physical partial differential equation (PDE) is important. We consider parabolic PDE mesh generation and present a method for the construction of adaptive meshes in two spatial dimensions using stochastic domain decomposition that is suitable for an implementation in a multi- or many-core environment. Methods for mesh generation on periodic domains are also provided. The mesh generator is coupled to a time dependent physical PDE and the system is evolved using an alternating solution procedure. The method uses the stochastic representation of the exact solution of a parabolic linear mesh generator to find the location of an adaptive mesh along the (artificial) subdomain interfaces. The deterministic evaluation of the mesh over each subdomain can then be obtained completely independently using the probabilistically computed solutions as boundary conditions. The parallel performance of this general stochastic domain decomposition approach has previously been shown. We demonstrate the approach numerically for the mesh generation context and compare the mesh obtained with the corresponding single domain mesh using a representative mesh quality measure.

Alexander Bihlo, Jean-Christophe Nave

Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy.