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A new 3-D numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes
- Source :
- Computer Methods in Applied Mechanics and Engineering. 354:568-592
- Publication Year :
- 2019
- Publisher :
- Elsevier BV, 2019.
-
Abstract
- A new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Laplace equation on irregular domains with the Dirichlet boundary conditions has been developed. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used for 3-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances ( 0 . 1 h − 1 0 − 9 h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fifth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.
- Subjects :
- Laplace's equation
Partial differential equation
Mechanical Engineering
Mathematical analysis
Computational Mechanics
General Physics and Astronomy
Order of accuracy
010103 numerical & computational mathematics
01 natural sciences
Stencil
Finite element method
Computer Science Applications
law.invention
010101 applied mathematics
symbols.namesake
Mechanics of Materials
law
Dirichlet boundary condition
symbols
Heat equation
Cartesian coordinate system
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 00457825
- Volume :
- 354
- Database :
- OpenAIRE
- Journal :
- Computer Methods in Applied Mechanics and Engineering
- Accession number :
- edsair.doi...........d94eddd31b4c990ef6ec8e4abda2e39e