TY - GEN
T1 - A new numerical approach to the solution of partial differential equations with optimal accuracy on irregular domains and cartesian meshes.
AU - Idesman, A.
N1 - Publisher Copyright:
Copyright © 2019 COMPDYN Proceedings. All rights reserved.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Cartesian meshes
KW - Heat
KW - Irregular domains
KW - Laplace equations
KW - Local truncation error
KW - Optimal accuracy
KW - Wave
UR - http://www.scopus.com/inward/record.url?scp=85079098399&partnerID=8YFLogxK
U2 - 10.7712/120119.7021.18485
DO - 10.7712/120119.7021.18485
M3 - Conference contribution
AN - SCOPUS:85079098399
T3 - COMPDYN Proceedings
SP - 1582
EP - 1611
BT - COMPDYN 2019 - 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Proceedings
A2 - Papadrakakis, Manolis
A2 - Fragiadakis, Michalis
PB - National Technical University of Athens
T2 - 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, COMPDYN 2019
Y2 - 24 June 2019 through 26 June 2019
ER -