Energy theorems, duality, and variational formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

Download Full-text

fenicsR13

ACM Transactions on Mathematical Software ◽

10.1145/3442378 ◽

2021 ◽

Vol 47 (2) ◽

pp. 1-29

Author(s):

Lambert Theisen ◽

Manuel Torrilhon

Keyword(s):

Gas Flow ◽

Differential Operators ◽

Mixed Finite Element ◽

Test Cases ◽

Typical Application ◽

Flow Models ◽

Computing Platform ◽

Abstraction Levels ◽

Variational Formulations ◽

Rarefaction Effects

We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.

Download Full-text

A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2020-0129 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Niklas Ericsson

Keyword(s):

Fourier Expansion ◽

Stokes Equations ◽

Stokes Problem ◽

Three Dimensional ◽

Countable Family ◽

Angular Variable ◽

Two Dimensional ◽

Sobolev Norms ◽

Well Posed ◽

Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Download Full-text