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Symmetries of Bernstein Polynomial Differentiation Matrices and Applications to Initial Value Problems

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Gepubliceerd in:Symmetry vol. 17, no. 1 (2025), p. 47
Hoofdauteur: Mirkov, Nikola
Andere auteurs: Fabiano, Nicola, Nikezić, Dušan, Stojiljković, Vuk, Ilić, Milica
Gepubliceerd in:
MDPI AG
Onderwerpen:
Differentiation
Accuracy
Partial differential equations
Mathematical analysis
Polynomials
Nodes
Collocation methods
Chebyshev approximation
Approximation
Numerical analysis
Diffusion rate
Methods
Stability
Numerical methods
Boundary value problems
Spectral methods
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Samenvatting:In this study, we discuss the symmetries underlying Bernstein polynomial differentiation matrices, as they are used in the collocation method approach to approximate solutions of initial and boundary value problems. The symmetries are brought into connection with those of the Chebyshev pseudospectral method (Chebyshev polynomial differentiation matrices). The treatment discussed here enables a faster and more accurate generation of differentiation matrices. The results are applied in numerical solutions of several initial value problems for the partial differential equation of convection–diffusion reaction type. The method described herein demonstrates the combination of advanced numerical techniques like polynomial interpolation, stability-preserving timestepping, and transformation methods to solve a challenging nonlinear PDE efficiently. The use of Bernstein polynomials offers a high degree of accuracy for spatial discretization, and the CGL nodes improve the stability of the polynomial approximation. This analysis shows that exploiting symmetry in the differentiation matrices, combined with the wise choice of collocation nodes (CGL), leads to both accurate and efficient numerical methods for solving PDEs and accuracy that approach pseudospectral methods that use well-known orthogonal polynomials such as Chebyshev polynomials.
ISSN:2073-8994
DOI:10.3390/sym17010047
Bron:Engineering Database