Invariant meshless discretization schemes

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.Comment: 10 pages, 2 figures, release versio

Precipitation nowcasting using a stochastic variational frame predictor with learned prior distribution

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.Comment: 7 pages, 3 figures, release versio

Symmetries in atmospheric sciences

Selected applications of symmetry methods in the atmospheric sciences are reviewed briefly. In particular, focus is put on the utilisation of the classical Lie symmetry approach to derive classes of exact solutions from atmospheric models. This is illustrated with the barotropic vorticity equation. Moreover, the possibility for construction of partially-invariant solutions is discussed for this model. A further point is a discussion of using symmetries for relating different classes of differential equations. This is illustrated with the spherical and the potential vorticity equation. Finally, discrete symmetries are used to derive the minimal finite-mode version of the vorticity equation first discussed by E. Lorenz (1960) in a sound mathematical fashion.Comment: 7 pages, conference proceeding

Group classification of linear evolution equations

The group classification problem for the class of (1+1)-dimensional linear $r$th order evolution equations is solved for arbitrary values of $r>2$. It is shown that a related maximally gauged class of homogeneous linear evolution equations is uniformly semi-normalized with respect to linear superposition of solutions and hence the complete group classification can be obtained using the algebraic method. We also compute exact solutions for equations from the class under consideration using Lie reduction and its specific generalizations for linear equations.Comment: Minor corrections, 24 pages, 1 tabl

Conservative parameterization schemes

Parameterization (closure) schemes in numerical weather and climate prediction models account for the effects of physical processes that cannot be resolved explicitly by these models. Methods for finding physical parameterization schemes that preserve conservation laws of systems of differential equations are introduced. These methods rest on the possibility to regard the problem of finding conservative parameterization schemes as a conservation law classification problem for classes of differential equations. The relevant classification problems can be solved using the direct or inverse classification procedures. In the direct approach, one starts with a general functional form of the parameterization scheme. Specific forms are then found so that corresponding closed equations admit conservation laws. In the inverse approach, one seeks parameterization schemes that preserve one or more pre-selected conservation laws of the initial model. The physical interpretation of both classification approaches is discussed. Special attention is paid to the problem of finding parameterization schemes that preserve both conservation laws and symmetries. All methods are illustrated by finding conservative and conservative invariant parameterization schemes for systems of one-dimensional shallow-water equations.Comment: 25 pages, release versio

Symmetry-preserving finite element schemes: An introductory investigation

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.Comment: 21 pages, 3 figure

Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures

A typical problem with the conventional Galerkin approach for the construction of finite-mode models is to keep structural properties unaffected in the process of discretization. We present two examples of finite-mode approximations that in some respect preserve the geometric attributes inherited from their continuous models: a three-component model of the barotropic vorticity equation known as Lorenz' maximum simplification equations [Tellus, \textbf{12}, 243--254 (1960)] and a six-component model of the two-dimensional Rayleigh--B\'{e}nard convection problem. It is reviewed that the Lorenz--1960 model respects both the maximal set of admitted point symmetries and an extension of the noncanonical Hamiltonian form (Nambu form). In a similar fashion, it is proved that the famous Lorenz--1963 model violates the structural properties of the Saltzman equations and hence cannot be considered as the maximum simplification of the Rayleigh--B\'{e}nard convection problem. Using a six-component truncation, we show that it is again possible retaining both symmetries and the Nambu representation in the course of discretization. The conservative part of this six-component reduction is related to the Lagrange top equations. Dissipation is incorporated using a metric tensor.Comment: 18 pages; extended and corrected version; added reference

Algebraic method for finding equivalence groups

The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the corresponding equivalence algebra. Two versions of the method are presented, where the first involves the automorphism group of this algebra and the second is based on a list of its megaideals. We illustrate the megaideal-based version of the method with the computation of the complete equivalence group of a class of nonlinear wave equations with applications in nonlinear elasticity.Comment: 17 pages; revised version; includes results that have been excluded from the journal version of the preprint arXiv:1106.4801v

Well-balanced mesh-based and meshless schemes for the shallow-water equations

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.Comment: 16 pages, 4 figure

Invariant parameterization and turbulence modeling on the beta-plane

Invariant parameterization schemes for the eddy-vorticity flux in the barotropic vorticity equation on the beta-plane are constructed and then applied to turbulence modeling. This construction is realized by the exhaustive description of differential invariants for the maximal Lie invariance pseudogroup of this equation using the method of moving frames, which includes finding functional bases of differential invariants of arbitrary order, a minimal generating set of differential invariants and a basis of operators of invariant differentiation in an explicit form. Special attention is paid to the problem of two-dimensional turbulence on the beta-plane. It is shown that classical hyperdiffusion as used to initiate the energy-enstrophy cascades violates the symmetries of the vorticity equation. Invariant but nonlinear hyperdiffusion-like terms of new types are introduced and then used in the course of numerically integrating the vorticity equation and carrying out freely decaying turbulence tests. It is found that the invariant hyperdiffusion scheme is close to but not exactly reproducing the 1/k shape of energy spectrum in the enstrophy inertial range. By presenting conservative invariant hyperdiffusion terms, we also demonstrate that the concepts of invariant and conservative parameterizations are consistent.Comment: 28 pages, 2 figures, revised and extended versio