1,318 research outputs found
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Eigenvalue problems are fundamental to mathematics and science. We present a
simple algorithm for determining eigenvalues and eigenfunctions of the
Laplace--Beltrami operator on rather general curved surfaces. Our algorithm,
which is based on the Closest Point Method, relies on an embedding of the
surface in a higher-dimensional space, where standard Cartesian finite
difference and interpolation schemes can be easily applied. We show that there
is a one-to-one correspondence between a problem defined in the embedding space
and the original surface problem. For open surfaces, we present a simple way to
impose Dirichlet and Neumann boundary conditions while maintaining second-order
accuracy. Convergence studies and a series of examples demonstrate the
effectiveness and generality of our approach
Strong Stability Preserving Two-Step Runge-Kutta Methods
We investigate the strong stability preserving (SSP) property of two-step Runge– Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple\ud
subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge–Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations
Strong stability preserving explicit Runge-Kutta methods of maximal effective order
We apply the concept of effective order to strong stability preserving (SSP)
explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods,
methods with an effective order of accuracy are designed to satisfy a relaxed
set of order conditions, but yield higher order accuracy when composed with
special starting and stopping methods. We show that this allows the
construction of four-stage SSP methods with effective order four (such methods
cannot have classical order four). However, we also prove that effective order
five methods - like classical order five methods - require the use of
non-positive weights and so cannot be SSP. By numerical optimization, we
construct explicit SSP Runge-Kutta methods up to effective order four and
establish the optimality of many of them. Numerical experiments demonstrate the
validity of these methods in practice.Comment: 17 pages, 3 figures, 8 table
Gossamer roadmap technology reference study for a solar polar mission
A technology reference study for a solar polar mission is presented. The study uses novel analytical methods to quantify the mission design space including the required sail performance to achieve a given solar polar observation angle within a given timeframe and thus to derive mass allocations for the remaining spacecraft sub-systems, that is excluding the solar sail sub-system. A parametric, bottom-up, system mass budget analysis is then used to establish the required sail technology to deliver a range of science payloads, and to establish where such payloads can be delivered to within a given timeframe. It is found that a solar polar mission requires a solar sail of side-length 100 – 125 m to deliver a ‘sufficient value’ minimum science payload, and that a 2. 5μm sail film substrate is typically required, however the design is much less sensitive to the boom specific mass
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