Schedule for: 17w5152 - Connections in Geometric Numerical Integration and Structure-Preserving Discretization

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will survey the key structures, examples, and applications of finite element exterior calculus (FEEC). FEEC is a canonical example of structure-preserving discretization which brings together geometry, topology, and analysis in order to unify, clarify, and refine many of the classical mixed finite element methods, and to enable the development of stable mixed finite elements for problems where they had previously not been available.

Many PDEs arise from a variational principle, and this leads to many of their qualitative properties. The discrete variational method (DVDM) is a technique for discretising such a PDE directly from this variational principle. It derivation ensures that it reproduces dissipation/energy preserving properties of the underlying PDE. The performance of these method is often very good. In this talk I will show that this is because the discrete solution satisfies a modified equation, which in turn satisfies a (form of a) variational principle which is a perturbation of the original. Properties of the solution of the DVDM can then be derived directly from this modified variational equation. This is Joint work with akaharu Yaguchi (Kobe) and Daisuke Furihata (Osaka).

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Time dependent nonlinear partial differential equations, like for example reaction diffusion equations, are usually solved by classical time marching schemes, like Runge-Kutta methods, or linear multi-step methods. Such equations can however have solutions which blow up in finite time, and in the blowup regime, the behavior of the solution is dominated by the non-linearity. I will show two different approaches how one can construct specialized numerical time integrators which take into account the physics of the underlying non-linear problem. I will show both theoretically and numerically that their performance can be orders of magnitude better than the performance of classical time integrators for such problems.

Among the major applications of discrete exterior calculus (in the sense of Hirani et al) are discretization of the Hodge-Laplace operator and various related problems. However, convergence issues for those problems are not completely resolved; as far as we are aware, there is no proof of convergence except for the Poisson equation in two dimensions, which is immediate because the discrete problem is identical to the one that arises from affine finite elements. Moreover, even in two dimensions, there have been some puzzling numerical experiments reported in the literature, apparently suggesting that there is convergence without consistency.

This talk discusses the application of hybridizable discontinuous Galerkin (HDG) methods to canonical Hamiltonian PDEs. We present necessary and sufficient conditions for an HDG method to satisfy a multisymplectic conservation law, when applied to such a system, and show that these conditions are satisfied by "hybridized" versions of several of the most commonly-used finite element methods. These finite element methods may therefore be used for high-order, structure-preserving discretization of Hamiltonian PDEs on unstructured meshes. (Joint work with Robert McLachlan.)

Given a differential equation that admits a group of symmetries, it is frequently desirable to preserve those symmetries when constructing a numerical scheme. For solutions that exhibit sharp variations or singularities, symmetry-preserving numerical schemes have proven to give very good numerical results. In the last 25 years, most of the research in this field has focused on the construction of symmetry-preserving finite difference numerical schemes. Extending known results to other types of numerical methods (such as finite element, finite volume, or spectral methods) remains a challenge. In this talk, I will report on preliminary results related to the construction of symmetry-preserving finite element numerical schemes. This is a joint work with Professor Alexander Bihlo from Memorial University.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Boris algorithm is the most popular time integrator for charged particle motion in electric and magnetic force fields. It is a symmetric one-step method, and it preserves the phase volume exactly. However, it is not symplectic. Nevertheless, numerical experiments confirm an excellent long-time near energy preservation of the system. In this talk we present a multistep extension of the Boris algorithm, which is explicit, symmetric, and has arbitrarily high order. Near preservation of energy and momentum for the underlying one-step method, and the boundedness of parasitic solution components are proved. A rigorous proof for the excellent near energy preservation of the Boris algorithm is still missing. (We thank Martin Gander for drawing our attention to this problem.)

We present a new class of conservative method, called the multiplier method, which enables systematic construction of conservative schemes for general dynamical systems. Specifically, the multiplier method can preserve arbitrary forms of conserved quantities and is applicable for systems without a symplectic or variational structure, such as dissipative problems. Moreover, we discuss a fundamental long-term stability property for general conservative methods. This is joint work with Alexander Bihlo and Jean-Christophe Nave.

B-series are a special form of Taylor series, where the terms are indexed by rooted trees. The theory originated by the seminal work on numerical integration by John Butcher half a century ago. The theory has developed to become arguably the most important tool in the study of structure preservation of numerical integrators. Geometrically B-series are intimately connected to the geometry of Euclidean spaces. Lie-Butcher series is a generalisation which combines B-series with Lie series, aimed at the study of flows on manifolds. In this talk we will discuss the interplay between the algebraic and geometric structures underlying Lie-Butcher series. We will review various recent results and work in progress, both for ODEs and in the theory of Rough Paths (stochastic DEs).

We solve the model of C.J. Budd, J.M. Stockie, Stud. Appl. Math. AA:1--29, 2015, treated numerically in the paper "Asymptotical computations for a model of flow in saturated porous media" by P. Amodio, C.J. Budd, O. Koch, G. Settanni and E. Weinmuller, Appl. Math. and Comput. 237 (2014), 155-167 by methods right out of Euler's "Institutiones Calculi Integralis" (1768/69/70). The obtained precision is only limited by that of the computer. A nice occasion to celebrate the 250th anniversary of this greatest classics for integration and differential equations.

In this talk we present a geometric variational discretization of equations describing compressible fluids. The numerical scheme is obtained by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in details the numerical scheme on 2D irregular simplicial meshes and evaluate the scheme numerically for the rotating shallow water equations. In particular, we investigate whether the scheme conserves stationary solutions, represents well the nonlinear dynamics, and approximates well the frequency relations of the continuous equations, while preserving conservation laws such as mass and total energy.

The problem is to combine explicit time and mimetic spatial discretizations for wave equations so that the energy is conserved. It is easy to see that intuitive discretizations of the energy are not conserved. However it is possible to find conserved quantities that depend on the time step that are conserved and converge quadratically in the time step to the classical energy.

The first half of this talk presents a general overview of Geometric Numerical Integration of differential equations. The second half discusses geometric integrability properties of Kahan¹s method.

Data assimilation methods are used for marrying instrumental observations of a physical system to numerical prediction models. There are many flavors, depending on whether one takes a probabilistic/statistical, control theoretic, or dynamical systems point of view. Furthermore there are variational methods that consider a whole time window and sequential methods that proceed step by-step. In this talk I will consider the relationship between the observation operator and the decomposition of the model tangent space in terms of Lyapunov exponents/vectors. the main conclusion is that the observations should constrain the unstable tangent space. Using this point of view we construct two methods, one variational and one sequential and illustrate their convergence behavior. Along the way I will mention some other structural considerations in data assimilation.

Meshless methods offer a flexible approach for problems with e.g. free surfaces, large boundary deformation, and Lagrangian dynamics. Historically, these methods have been unable to simultaneously maintain discrete conservation and high-order accuracy. This is because the lack of a mesh leads to a lack of a Stokes theorem, precluding the use of traditional tools when developing new schemes. We present several new results demonstrating how generalized moving least squares may be used to achieve compatible finite element-like results in a meshless setting. We demonstrate schemes for the div-grad, curl-curl and grad-div elliptic problems and use them to develop a high-order discretization for Stokes flow that we use to consider problems in dense electrophoretic suspensions. This is joint work with Mauro Perego and Pavel Bochev.

Projection operators which commute with the governing differential operators are key tools for the stability analysis of finite element methods associated to a differential complex. In fact, such projections have been a central feature of the analysis of mixed finite element methods since the beginning of such analysis. However, a key difficulty is that, for most of the standard finite element spaces, the canonical projection operators based on the degrees of freedom require additional smoothness to be well--defined and thus are not bounded on the appropriate function spaces. More recently, bounded commuting projections have been constructed, but these lack a key property of the canonical projections; they are not locally defined. In this talk, we review the ideas behind the construction of bounded projections that commute with the exterior derivative and show how, using local operators defined on overlapping macroelements, it is possible to construct such operators that are also locally defined.

When we want to use reference points located arbitrarily in two- or three-dimensional regions, it is essentially difficult to design some structure-preserving methods. The reason is that we should discretize some Gauss-Green formulae keeping some mathematical properties in that situations. Based on Voronoi-Delaunay triangulations, we can find some beautiful discrete Gauss-Green formulae and apply them to design some structure-preserving numerical methods. In the talk, we will indicate those formulae and their proofs in detail and the obtained discrete variational derivative methods based on Voronoi cells. Furthermore, if it is possible, we will show some relaxed structure-preserving methods to decrease computation cost.

In an effort to better understand what makes a numerical method mimetic we consider an exceptional case. The Keller-Box scheme is multi-symplectic (Reich 2000), it always propagates waves in the correct direction (Frank, 2006), and it discretizes the problem physics and calculus exactly (mimetic). However, the method is not easily described by algebraic topology or discrete differential forms. The properties of this unusual mimetic method are discussed and compared to the more classical mimetic finite element and finite volume methods.

Instead of the usual presentation given to the formulation of Initial Boundary Boundary Value Problems (IBVP), we do no take the partition of the continuous media directly to the limit of zero shrinking size concerning the spatial dimensions at any given time, which leads to some differential expression for the limiting net force upon the element. We consider each media element as a single particle evolving in “time” under a “force” represented by the discrete mimetic analog of the differential expression. We base our approach on a discrete extended Gauss’s divergence theorem, without using exterior calculus, to construct our mimetic operators combined with a symplectic integration scheme.

Variational integrators for Lagrangian systems have the advantage of conserving the momenta up to machine precision, independent of the time step. While the theory of variational integrators for mechanical systems is well developed, there applications of these integrators to systems involving fluid-structure interactions have proven difficult. In this talk, we derive a variational integrator for a particular type of fluid-structure interactions, namely, simulating the dynamics of a bendable tube conveying ideal fluid that can change its cross-section (collapsible tube). First, we derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluid’s back-to-labels map, coupled with a variational discretization of elastic part of the Lagrangian. We also show the results of simulations of the system with the spatial discretization using a very small number of points demonstrating a non-trivial and interesting behavior.

Integral preserving schemes for ODEs can be derived by means of for instance discrete gradient methods. For PDEs, one may first discretize in space, using for instance finite difference methods or finite element methods, and then apply an integral preserving method for the corresponding ODEs. For PDEs discretized on moving grids, the situation is more complicated, it is not even clear exactly what should be meant by an integral preserving scheme in this setting. We shall propose a definition and then derive the resulting conservative schemes, both with finite difference schemes and with finite element schemes. We test the methods on problems with travelling wave solutions and demonstrate that they give remarkably good results, both compared to fixed grid and to non-conservative schemes. This is joint work with S. Eidnes and T. Ringholm.

The strengths of the schemes we will present are their high order of accuracy in both space and time combined with their ability to march in time with a time step at the domain of dependence limit independent of the order. Additionally, the methods are globally super-convergent, i.e. the number of degrees of freedom per cell is (m+1)^d but the methods achieve orders of accuracy 2m. We note that the L2 super-convergence holds globally in space and time, unlike most other spatial discretizations, where super-convergence is limited to a few specific points and often rely on the use of negative norms. Our primary interest of these schemes are as highly efficient building blocks in hybrid methods where most of the mesh can be taken to be rectilinear and where geometry is handled by more flexible (but less efficient) methods close to physical boundaries. In this work we restrict our consideration to square geometries with boundary conditions of Dirichlet, Neumann or periodic type. We provide stability and convergence results for one dimensional periodic domains. The analysis of the conservative method is quite different from the analysis of previous dissipative Hermite methods and introduces a, to our knowledge, novel technique for analyzing conservative schemes for wave equations in second order form. This is joint work with Thomas Hagstrom (SMU) and Arturo Vargas (Rice, LLNL)

Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. Shapes are in fact unparametrized curves, evolving on a vector space, on a Lie group or on a manifold. One popular approach to shape analysis is by the use of the Square Root Velocity Transform (SRVT). In this talk we propose a generalisation of the SRVT from vector spaces to Lie groups and to homogeneous manifolds. This is Joint work with S. Eidnes, A. Schmeding.

In this work the High Order Mimetic Discretization Framework will be presented together with a discussion of two fundamental aspects for the construction of structure-preserving discretizations: (i) the definition of the discrete degrees of freedom of physical field quantities, and (ii) the formulation of the physical field laws. For the first, the geometric degrees of freedom (associated to point, lines, surfaces and volumes) will be introduced. For the second, the Navier Stokes equations will be used as an example.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.