45,583 research outputs found
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
Port-Hamiltonian discretization for open channel flows
A finite-dimensional Port-Hamiltonian formulation for the dynamics of smooth open channel flows is presented. A numerical scheme based on this formulation is developed for both the linear and nonlinear shallow water equations. The scheme is verified against exact solutions and has the advantage of conservation of mass and energy to the discrete level
High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows
In this article we present the first better than second order accurate
unstructured Lagrangian-type one-step WENO finite volume scheme for the
solution of hyperbolic partial differential equations with non-conservative
products. The method achieves high order of accuracy in space together with
essentially non-oscillatory behavior using a nonlinear WENO reconstruction
operator on unstructured triangular meshes. High order accuracy in time is
obtained via a local Lagrangian space-time Galerkin predictor method that
evolves the spatial reconstruction polynomials in time within each element. The
final one-step finite volume scheme is derived by integration over a moving
space-time control volume, where the non-conservative products are treated by a
path-conservative approach that defines the jump terms on the element
boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian
(ALE) method, where the mesh velocity can be chosen independently of the fluid
velocity.
The new scheme is applied to the full seven-equation Baer-Nunziato model of
compressible multi-phase flows in two space dimensions. The use of a Lagrangian
approach allows an excellent resolution of the solid contact and the resolution
of jumps in the volume fraction. The high order of accuracy of the scheme in
space and time is confirmed via a numerical convergence study. Finally, the
proposed method is also applied to a reduced version of the compressible
Baer-Nunziato model for the simulation of free surface water waves in moving
domains. In particular, the phenomenon of sloshing is studied in a moving water
tank and comparisons with experimental data are provided
Efficient computation of two-dimensional steady free-surface flows
We consider a family of steady free-surface flow problems in two dimensions,
concentrating on the effect of nonlinearity on the train of gravity waves that
appear downstream of a disturbance. By exploiting standard complex variable
techniques, these problems are formulated in terms of a coupled system of
Bernoulli's equation and an integral equation. When applying a numerical
collocation scheme, the Jacobian for the system is dense, as the integral
equation forces each of the algebraic equations to depend on each of the
unknowns. We present here a strategy for overcoming this challenge, which leads
to a numerical scheme that is much more efficient than what is normally
employed for these types of problems, allowing for many more grid points over
the free surface. In particular, we provide a simple recipe for constructing a
sparse approximation to the Jacobian that is used as a preconditioner in a
Jacobian-free Newton-Krylov method for solving the nonlinear system. We use
this approach to compute numerical results for a variety of prototype problems
including flows past pressure distributions, a surface-piercing object and
bottom topographies.Comment: 20 pages, 13 figures, under revie
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