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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.05598 |
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| _version_ | 1866911092478836736 |
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| author | Amaro, Mário B. |
| author_facet | Amaro, Mário B. |
| contents | We describe a short, reproducible workflow for applying finite differences on nonuniform grids determined by a positive weight function g. The grid is obtained by equidistribution, mapping uniform computational coordinates $ξ\in[0,1]$ to physical space by the cumulative integral $S(x)=\int_a^x\!1/g(s)\,ds$ and its inverse, and in multiple dimensions by the corresponding variable-diffusion (harmonic) mapping with tensor $P=(1/g)I$. We then use the standard three-point central stencils on uneven spacing for first and second derivatives. We collect the formulas, state the mild constraints on g (positivity, boundedness, integrability), and provide a small reference implementation. Finally, we solve the 2D time-independent Schrödinger equation for a harmonic oscillator on uniform vs. variable meshes, showing the expected improvement in resolving localized eigenfunctions without increasing matrix size. We intend this note as a how-to reference rather than a new method, consolidating widely used ideas into a single, ready-to-use recipe, claiming no novelty. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05598 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A practical recipe for variable-step finite differences via equidistribution Amaro, Mário B. Numerical Analysis Computational Physics We describe a short, reproducible workflow for applying finite differences on nonuniform grids determined by a positive weight function g. The grid is obtained by equidistribution, mapping uniform computational coordinates $ξ\in[0,1]$ to physical space by the cumulative integral $S(x)=\int_a^x\!1/g(s)\,ds$ and its inverse, and in multiple dimensions by the corresponding variable-diffusion (harmonic) mapping with tensor $P=(1/g)I$. We then use the standard three-point central stencils on uneven spacing for first and second derivatives. We collect the formulas, state the mild constraints on g (positivity, boundedness, integrability), and provide a small reference implementation. Finally, we solve the 2D time-independent Schrödinger equation for a harmonic oscillator on uniform vs. variable meshes, showing the expected improvement in resolving localized eigenfunctions without increasing matrix size. We intend this note as a how-to reference rather than a new method, consolidating widely used ideas into a single, ready-to-use recipe, claiming no novelty. |
| title | A practical recipe for variable-step finite differences via equidistribution |
| topic | Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2412.05598 |