38 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
Variable step-size implicit-explicit linear multistep methods for time-dependent partial differential equations
Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving
partial differential equations (PDEs) with terms of different types. While fixed time-step versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit (VSIMEX) linear multistep methods for time-dependent PDEs. Families of order-p, p-step VSIMEX schemes are constructed and analyzed, where
p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property
that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers’ equation. These results demonstrate that by varying the time
step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.The work of the first and second authors was partially supported by an NSERC Canada Postgraduate Scholarship, and a grant from NSERC Canada, respectively.published or submitted for publicationis peer reviewe
Implicit-explicit methods for time-dependent PDE’s
Various 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