1,571 research outputs found
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
The closest point method (Ruuth and Merriman, J. Comput. Phys.
227(3):1943-1961, [2008]) is an embedding method developed to solve a variety
of partial differential equations (PDEs) on smooth surfaces, using a closest
point representation of the surface and standard Cartesian grid methods in the
embedding space. Recently, a closest point method with explicit time-stepping
was proposed that uses finite differences derived from radial basis functions
(RBF-FD). Here, we propose a least-squares implicit formulation of the closest
point method to impose the constant-along-normal extension of the solution on
the surface into the embedding space. Our proposed method is particularly
flexible with respect to the choice of the computational grid in the embedding
space. In particular, we may compute over a computational tube that contains
problematic nodes. This fact enables us to combine the proposed method with the
grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024,
[2009]) to obtain a numerical method for approximating PDEs on moving surfaces.
We present a number of examples to illustrate the numerical convergence
properties of our proposed method. Experiments for advection-diffusion
equations and Cahn-Hilliard equations that are strongly coupled to the velocity
of the surface are also presented
Weak Continuity and Compactness for Nonlinear Partial Differential Equations
We present several examples of fundamental problems involving weak continuity
and compactness for nonlinear partial differential equations, in which
compensated compactness and related ideas have played a significant role. We
first focus on the compactness and convergence of vanishing viscosity solutions
for nonlinear hyperbolic conservation laws, including the inviscid limit from
the Navier-Stokes equations to the Euler equations for homentropy flow, the
vanishing viscosity method to construct the global spherically symmetric
solutions to the multidimensional compressible Euler equations, and the
sonic-subsonic limit of solutions of the full Euler equations for
multidimensional steady compressible fluids. We then analyze the weak
continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding
isometric embeddings in differential geometry. Further references are also
provided for some recent developments on the weak continuity and compactness
for nonlinear partial differential equations.Comment: 29 page
A finite element approach for vector- and tensor-valued surface PDEs
We derive a Cartesian componentwise description of the covariant derivative
of tangential tensor fields of any degree on general manifolds. This allows to
reformulate any vector- and tensor-valued surface PDE in a form suitable to be
solved by established tools for scalar-valued surface PDEs. We consider
piecewise linear Lagrange surface finite elements on triangulated surfaces and
validate the approach by a vector- and a tensor-valued surface Helmholtz
problem on an ellipsoid. We experimentally show optimal (linear) order of
convergence for these problems. The full functionality is demonstrated by
solving a surface Landau-de Gennes problem on the Stanford bunny. All tools
required to apply this approach to other vector- and tensor-valued surface PDEs
are provided
A Meshfree Generalized Finite Difference Method for Surface PDEs
In this paper, we propose a novel meshfree Generalized Finite Difference
Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative
approximations for the same are done directly on the tangent space, in a manner
that mimics the procedure followed in volume-based meshfree GFDMs. As a result,
the proposed method not only does not require a mesh, it also does not require
an explicit reconstruction of the manifold. In contrast to existing methods, it
avoids the complexities of dealing with a manifold metric, while also avoiding
the need to solve a PDE in the embedding space. A major advantage of this
method is that all developments in usual volume-based numerical methods can be
directly ported over to surfaces using this framework. We propose
discretizations of the surface gradient operator, the surface Laplacian and
surface Diffusion operators. Possibilities to deal with anisotropic and
discontinous surface properties (with large jumps) are also introduced, and a
few practical applications are presented
- …