1,236 research outputs found
PDE-Based Multidimensional Extrapolation of Scalar Fields over Interfaces with Kinks and High Curvatures
We present a PDE-based approach for the multidimensional extrapolation of
smooth scalar quantities across interfaces with kinks and regions of high
curvature. Unlike the commonly used method of [2] in which normal derivatives
are extrapolated, the proposed approach is based on the extrapolation and
weighting of Cartesian derivatives. As a result, second- and third-order
accurate extensions in the norm are obtained with linear and
quadratic extrapolations, respectively, even in the presence of sharp geometric
features. The accuracy of the method is demonstrated on a number of examples in
two and three spatial dimensions and compared to the approach of [2]. The
importance of accurate extrapolation near sharp geometric features is
highlighted on an example of solving the diffusion equation on evolving
domains.Comment: 17 pages, 13 figures, submitted to SIAM Journal of Scientific
Computin
Decoupling Numerical Method Based on Deep Neural Network for Nonlinear Degenerate Interface Problems
Interface problems depict many fundamental physical phenomena and widely
apply in the engineering. However, it is challenging to develop efficient fully
decoupled numerical methods for solving degenerate interface problems in which
the coefficient of a PDE is discontinuous and greater than or equal to zero on
the interface. The main motivation in this paper is to construct fully
decoupled numerical methods for solving nonlinear degenerate interface problems
with ``double singularities". An efficient fully decoupled numerical method is
proposed for nonlinear degenerate interface problems. The scheme combines deep
neural network on the singular subdomain with finite difference method on the
regular subdomain. The key of the new approach is to split nonlinear degenerate
partial differential equation with interface into two independent boundary
value problems based on deep learning. The outstanding advantages of the
proposed schemes are that not only the convergence order of the degenerate
interface problems on whole domain is determined by the finite difference
scheme on the regular subdomain, but also can calculate
jump ratio(such as or ) for the interface
problems including degenerate and non-degenerate cases. The expansion of the
solutions does not contains any undetermined parameters in the numerical
method. In this way, two independent nonlinear systems are constructed in other
subdomains and can be computed in parallel. The flexibility, accuracy and
efficiency of the methods are validated from various experiments in both 1D and
2D. Specially, not only our method is suitable for solving degenerate interface
case, but also for non-degenerate interface case. Some application examples
with complicated multi-connected and sharp edge interface examples including
degenerate and nondegenerate cases are also presented
A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I
Physics informed neural networks (PINNs) have emerged as a powerful tool to
provide robust and accurate approximations of solutions to partial differential
equations (PDEs). However, PINNs face serious difficulties and challenges when
trying to approximate PDEs with dominant hyperbolic character. This research
focuses on the development of a physics informed deep learning framework to
approximate solutions to nonlinear PDEs that can develop shocks or
discontinuities without any a-priori knowledge of the solution or the location
of the discontinuities. The work takes motivation from finite element method
that solves for solution values at nodes in the discretized domain and use
these nodal values to obtain a globally defined solution field. Built on the
rigorous mathematical foundations of the discontinuous Galerkin method, the
framework naturally handles imposition of boundary conditions
(Neumann/Dirichlet), entropy conditions, and regularity requirements. Several
numerical experiments and validation with analytical solutions demonstrate the
accuracy, robustness, and effectiveness of the proposed framework
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Advanced Computational Engineering
The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications
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