A new numerical approach to the solution of partial differential equations with optimal accuracy on irregular domains and cartesian meshes.

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Abstract

A new numerical approach for the time dependent wave and heat equations as well as for the time independent Laplace equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-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. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-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 also show that at the same number of degrees of freedom, the new approach even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Laplace equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

Original languageEnglish
Title of host publicationCOMPDYN 2019 - 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Proceedings
EditorsManolis Papadrakakis, Michalis Fragiadakis
PublisherNational Technical University of Athens
Pages1582-1611
Number of pages30
ISBN (Electronic)9786188284463
DOIs
StatePublished - 2019
Event7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, COMPDYN 2019 - Crete, Greece
Duration: Jun 24 2019Jun 26 2019

Publication series

NameCOMPDYN Proceedings
Volume1
ISSN (Print)2623-3347

Conference

Conference7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, COMPDYN 2019
CountryGreece
CityCrete
Period06/24/1906/26/19

Keywords

  • Cartesian meshes
  • Heat
  • Irregular domains
  • Laplace equations
  • Local truncation error
  • Optimal accuracy
  • Wave

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