39 research outputs found
A Diffusion-Generated Approach to Multiphase Motion
No full textIn this article, we present a diffusion-generated approach for evolving multiple junctions. This work generalizes an earlier method by Merriman, Bence and Osher which alternately diffuses and sharpens characteristic functions for each phase region to produce pure mean curvature flow. Specifically, our new method produces a normal velocity equal to a positive multiple of the curvature of the interface plus the difference in bulk energies for prescribed junction angles. This simple method naturally treats topological mergings and breakings, produces no overlapping regions or vacuums and can be made very fast. Numerical studies are provided which show that our method agrees with front tracking and a recent variational approach for a variety of examples. Asymptotic expansions are also carried out near junctions to justify our algorithms. Department of Mathematics, University of California at Los Angeles. ([email protected]). The work of this author was partially supported by an NSERC P..
A Fixed Grid Method for Capturing the Motion of Self-Intersecting Interfaces and Related PDEs
No full textMoving interfaces that self-intersect arise naturally in the geometric optics model of wavefront motion. Ray tracing techniques can be used to compute these motions, but they lose resolution as rays diverge. In this paper we develop a new numerical method that maintains uniform spatial resolution of the front at all times. Our approach is a fixed grid, interface capturing formulation based on the Dynamic Surface Extension method of Steinhoff and Fan [10]. The new methods can treat arbitrarily complicated self intersecting fronts, as well as refraction, reflection and focusing. We also further extend this approach to curvature dependent front motions, and the motion of filaments. We validate the new methods with numerical experiments. Department of Mathematics, University of California at Los Angeles. ([email protected]). The work of this author was partially supported by AFOSR STTR FQ8671-9801346. y Department of Mathematics, University of California at Los Angeles. ([email protected]..
An embedding technique for the solution of reaction–diffusion equations on algebraic surfaces with isolated singularities
No full textIn this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities.The authors thank Colin Macdonald for many helpful discussions. The authors also thank the people who contributed to the cp_matrices code (see github.com/cbm755/cp_matrices), in particular Colin Macdonald, Ingrid von Glehn, and Yujia Chen. Thomas März was supported by award KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST). Parousia Rockstroh and Steven Ruuth were supported by a grant from NSERC Canada (RGPIN 227823) and by award KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST)
Diffusion-Generated Motion by Mean Curvature for Filaments
No full textIn previous work, we introduced a simple algorithm for producing motion by mean curvature of a surface, in which the motion is generated by alternately diffusing and renormalizing a characteristic function. This procedure is interesting due to its extreme simplicity, and because it isolates the connection between diffusion and curvature motion. However, it makes sense only for surfaces, i.e. objects of dimension d \Gamma 1 inside of R d . In this paper, we generalize diffusion-generated motion to a procedure that can be applied to objects of any dimension k inside of R d , k ! d. We focus on generating the curvature motion of filaments, i.e. curves in R 3 , since this is an important special case which also illustrates the general ideas. The method for filaments consists of applying diffusion to a complex valued function whose values wind Department of Mathematics, University of California at Los Angeles. ([email protected]). The work of this author was partially supported ..
On the Linear Stability of the Fifth-Order WENO Discretization
No full textWe study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge-Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge-Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis. © Springer Science+Business Media, LLC 2010.The work of M. Motamed was partially supported by NSERC Canada.The work of C. B. Macdonald was supported by NSERC Canada, NSF grant number CCF-0321917, and by Award No KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST).The work of S.J. Ruuth was partially supported by NSERC Canada
Efficient Algorithms for Diffusion-Generated Motion by Mean Curvature
No full textThe problem of simulating the motion of evolving surfaces with junctions according to some curvature-dependent speed arises in a number of applications. By alternately diffusing and sharpening characteristic functions for each region, a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums. However, the usual finite difference discretization of these methods are often excessively slow when accurate solutions are sought, even in two dimensions. We propose a new, spectral discretization of these diffusion-generated methods which obtains greatly improved efficiency over the usual finite difference approach. These efficiency gains are obtained, in part, through the use of a quadrature-based refinement technique, by integrating Fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients. Indeed, numerical studies demonstrate that the new algorithm is often more than 1000 ti..
GLOBAL OPTIMIZATION OF EXPLICIT STRONG-STABILITY-PRESERVING RUNGE-KUTTA METHODS
No full textAbstract. Strong-stability-preserving Runge-Kutta (SSPRK) methods are a type of time discretization method that are widely used, especially for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE; e.g., positivity or stability with respect to total variation. This is of particular interest when the solution exhibits shock-like or other nonsmooth behaviour. A variety of optimality results have been proven for simple SSPRK methods. However, the scope of these results has been limited to low-order methods due to the detailed nature of the proofs. In this article, global optimization software, BARON, is applied to an appropriate mathematical formulation to obtain optimality results for general explicit SSPRK methods up to fifth-order and explicit low-storage SSPRK methods up to fourth-order. Throughout, our studies allow for the possibility of negative coefficients which correspond to downwind-biased spatial discretizations. Guarantees of optimality are obtained for a variety of third- and fourth-order schemes. Where optimality is impractical to guarantee (specifically, for fifthorder methods and certain low-storage methods), extensive numerical optimizations are carried out to derive numerically optimal schemes. As a part of these studies, several new schemes arise which have theoretically improved time-stepping restrictions over schemes appearing in the recent literature. 1
Implicit-explicit methods for time-dependent PDE’s
No full textVarious methods have been proposed to integrate dynamical systems arising from spatially discretized time-dependent partial differential equations. For problems with terms of different types, implicit-explicit (IMEX) schemes have been used, especially in conjunction with spectral methods. For convection-diffusion problems, for example, one would use an explicit scheme for the convection term and an implicit scheme for thediffusion term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and aliasing reduction for spectral methods.
For the prototype linear advection-diffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified.
Numerical experiments demonstrate that weak decay of high frequency modes can lead to extra iterations on the finest grid when using multigrid computations with finite difference spatial discretization, and to aliasing when using spectral collocation for spatial discretization. When this behaviour occurs, use of weakly damping schemes such as the popular combination of Crank-Nicolson with second order Adams-Bashforth is discouraged and better alternatives are proposed.
Our findings are demonstrated on several examples.Science, Faculty ofMathematics, Department ofGraduat
Implicit-Explicit Methods For Reaction-Diffusion Problems In Pattern Formation
No full text. Reaction-diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems. To approximate the corresponding spatially discretized models, an explicit scheme can be used for the reaction term and an implicit scheme for the diffusion term. Such implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods in fluid flow problems. In this work, we analyze the performance of several of the best known linear multistep IMEX schemes for reaction-diffusion problems in pattern formation. For the linearized two chemical system, the growth properties exhibited by IMEX schemes are examined. Schemes which accurately represent the growth of the linearized problem for large time steps are identified. Numerical experiments show that first order accurate schemes, as well as schemes which produce only a weak decay of high frequency spatial error may yield plausible results which are nonetheless quali..
