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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.04142 |
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| _version_ | 1866915141584420864 |
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| author | Lewis, T. Xue, X. |
| author_facet | Lewis, T. Xue, X. |
| contents | Linear stationary reaction-convection-diffusion equations with Dirichlet boundary conditions are approximated using a simple finite difference method corresponding to central differences and the addition of a high-order stabilization term called a numerical moment. The focus is on convection-dominated equations, and the formulation for the method is motivated by various results for fully nonlinear problems. The method features higher-order local truncation errors than monotone methods consistent with the use of the central difference approximation for the gradient. Stability and rates of convergence are derived in the $\ell^2$ norm for the constant-coefficient case. Numerical tests are provided to compare the new methods to monotone methods. The methods are also tested for stationary Hamilton-Jacobi equations where they demonstrate higher rates of convergence than the Lax-Friedrich's method when the underlying viscosity solution is smooth and comparable performance when the underlying viscosity solution is not smooth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04142 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical moment stabilization of central difference approximations for linear stationary reaction-convection-diffusion equations with applications to stationary Hamilton-Jacobi equations Lewis, T. Xue, X. Numerical Analysis 65N06, 65N12 Linear stationary reaction-convection-diffusion equations with Dirichlet boundary conditions are approximated using a simple finite difference method corresponding to central differences and the addition of a high-order stabilization term called a numerical moment. The focus is on convection-dominated equations, and the formulation for the method is motivated by various results for fully nonlinear problems. The method features higher-order local truncation errors than monotone methods consistent with the use of the central difference approximation for the gradient. Stability and rates of convergence are derived in the $\ell^2$ norm for the constant-coefficient case. Numerical tests are provided to compare the new methods to monotone methods. The methods are also tested for stationary Hamilton-Jacobi equations where they demonstrate higher rates of convergence than the Lax-Friedrich's method when the underlying viscosity solution is smooth and comparable performance when the underlying viscosity solution is not smooth. |
| title | Numerical moment stabilization of central difference approximations for linear stationary reaction-convection-diffusion equations with applications to stationary Hamilton-Jacobi equations |
| topic | Numerical Analysis 65N06, 65N12 |
| url | https://arxiv.org/abs/2502.04142 |