87 research outputs found
Discrete conservation properties for shallow water flows using mixed mimetic spectral elements
A mixed mimetic spectral element method is applied to solve the rotating
shallow water equations. The mixed method uses the recently developed spectral
element histopolation functions, which exactly satisfy the fundamental theorem
of calculus with respect to the standard Lagrange basis functions in one
dimension. These are used to construct tensor product solution spaces which
satisfy the generalized Stokes theorem, as well as the annihilation of the
gradient operator by the curl and the curl by the divergence. This allows for
the exact conservation of first order moments (mass, vorticity), as well as
quadratic moments (energy, potential enstrophy), subject to the truncation
error of the time stepping scheme. The continuity equation is solved in the
strong form, such that mass conservation holds point wise, while the momentum
equation is solved in the weak form such that vorticity is globally conserved.
While mass, vorticity and energy conservation hold for any quadrature rule,
potential enstrophy conservation is dependent on exact spatial integration. The
method possesses a weak form statement of geostrophic balance due to the
compatible nature of the solution spaces and arbitrarily high order spatial
error convergence
Mimetic Spectral Element advection
We present a discretization of the linear advection of differential forms on
bounded domains. The framework previously established is extended to
incorporate the Lie derivative, , by means of Cartan's homotopy
formula. The method is based on a physics-compatible discretization with
spectral accuracy . It will be shown that the derived scheme has spectral
convergence with local mass conservation. Artificial dispersion depends on the
order of time integration
Triangular spectral elements for incompressible fluid flow
We discuss the use of triangular elements in the spectral element method for direct simulation of incompressible flow. Triangles provide much greater geometric flexibility than quadrilateral elements and are better conditioned and more accurate when small angles arise. We employ a family of tensor product algorithms for triangles, allowing triangular elements to be handled with comparable arithmetic complexity to quadrilateral elements. The triangular discretizations are applied and validated on the Poisson equation. These discretizations are then applied to the incompressible Navier-Stokes equations and a laminar channel flow solution is given. These new triangular spectral elements can be combined with standard quadrilateral elements, yielding a general and flexible high order method for complex geometries in two dimensions
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
MultiShape: A Spectral Element Method, with Applications to Dynamic Density Functional Theory and PDE-Constrained Optimization
A numerical framework is developed to solve various types of PDEs on
complicated domains, including steady and time-dependent, non-linear and
non-local PDEs, with different boundary conditions that can also include
non-linear and non-local terms. This numerical framework, called MultiShape, is
a class in Matlab, and the software is open source. We demonstrate that
MultiShape is compatible with other numerical methods, such as
differential--algebraic equation solvers and optimization algorithms. The
numerical implementation is designed to be user-friendly, with most of the
set-up and computations done automatically by MultiShape and with intuitive
operator definition, notation, and user-interface. Validation tests are
presented, before we introduce three examples motivated by applications in
Dynamic Density Functional Theory and PDE-constrained optimization,
illustrating the versatility of the method
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