123 research outputs found
Tracking bifurcating solutions of a model biological pattern generator
We study heterogeneous steady-state solutions of a cell-chemotaxis model for generating biological spatial patterns in two-dimensional domains with zero flux boundary conditions. We use the finite-element package ENTWIFE to investigate bifurcation from the uniform solution as the chemotactic parameter varies and as the domain scale and geometry change. We show that this simple cell-chemotaxis model can produce a remarkably wide and surprising range of complex spatial patterns
An accurate, fast, mathematically robust, universal, non-iterative algorithm for computing multi-component diffusion velocities
Using accurate multi-component diffusion treatment in numerical combustion
studies remains formidable due to the computational cost associated with
solving for diffusion velocities. To obtain the diffusion velocities, for low
density gases, one needs to solve the Stefan-Maxwell equations along with the
zero diffusion flux criteria, which scales as , when solved
exactly. In this article, we propose an accurate, fast, direct and robust
algorithm to compute multi-component diffusion velocities. To our knowledge,
this is the first provably accurate algorithm (the solution can be obtained up
to an arbitrary degree of precision) scaling at a computational complexity of
in finite precision. The key idea involves leveraging the fact
that the matrix of the reciprocal of the binary diffusivities, , is low
rank, with its rank being independent of the number of species involved. The
low rank representation of matrix is computed in a fast manner at a
computational complexity of and the Sherman-Morrison-Woodbury
formula is used to solve for the diffusion velocities at a computational
complexity of . Rigorous proofs and numerical benchmarks
illustrate the low rank property of the matrix and scaling of the
algorithm.Comment: 16 pages, 7 figures, 1 table, 1 algorith
An efficient algorithm for the parallel solution of high-dimensional differential equations
The study of high-dimensional differential equations is challenging and
difficult due to the analytical and computational intractability. Here, we
improve the speed of waveform relaxation (WR), a method to simulate
high-dimensional differential-algebraic equations. This new method termed
adaptive waveform relaxation (AWR) is tested on a communication network
example. Further we propose different heuristics for computing graph partitions
tailored to adaptive waveform relaxation. We find that AWR coupled with
appropriate graph partitioning methods provides a speedup by a factor between 3
and 16
An advancing front Delaunay triangulation algorithm designed for robustness
A new algorithm is described for generating an unstructured mesh about an arbitrary two-dimensional configuration. Mesh points are generated automatically by the algorithm in a manner which ensures a smooth variation of elements, and the resulting triangulation constitutes the Delaunay triangulation of these points. The algorithm combines the mathematical elegance and efficiency of Delaunay triangulation algorithms with the desirable point placement features, boundary integrity, and robustness traditionally associated with advancing-front-type mesh generation strategies. The method offers increased robustness over previous algorithms in that it cannot fail regardless of the initial boundary point distribution and the prescribed cell size distribution throughout the flow-field
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Common transversals and tangents to two lines and two quadrics in P^3
We solve the following geometric problem, which arises in several
three-dimensional applications in computational geometry: For which
arrangements of two lines and two spheres in R^3 are there infinitely many
lines simultaneously transversal to the two lines and tangent to the two
spheres?
We also treat a generalization of this problem to projective quadrics:
Replacing the spheres in R^3 by quadrics in projective space P^3, and fixing
the lines and one general quadric, we give the following complete geometric
description of the set of (second) quadrics for which the 2 lines and 2
quadrics have infinitely many transversals and tangents: In the
nine-dimensional projective space P^9 of quadrics, this is a curve of degree 24
consisting of 12 plane conics, a remarkably reducible variety.Comment: 26 pages, 9 .eps figures, web page with more pictures and and archive
of computations: http://www.math.umass.edu/~sottile/pages/2l2s
Unstructured mesh algorithms for aerodynamic calculations
The use of unstructured mesh techniques for solving complex aerodynamic flows is discussed. The principle advantages of unstructured mesh strategies, as they relate to complex geometries, adaptive meshing capabilities, and parallel processing are emphasized. The various aspects required for the efficient and accurate solution of aerodynamic flows are addressed. These include mesh generation, mesh adaptivity, solution algorithms, convergence acceleration, and turbulence modeling. Computations of viscous turbulent two-dimensional flows and inviscid three-dimensional flows about complex configurations are demonstrated. Remaining obstacles and directions for future research are also outlined
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