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A Hybrid Semi-Lagrangian Cut Cell Method for Advection-Diffusion Problems with Robin Boundary Conditions in Moving Domains
Numerical Analysis · Submitted 16 Feb 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We demonstrate second order accuracy in the $L^1$, $L^2$, and $L^\infty$ norms for both analytic test problems and numerical convergence studies.
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The abstract, sentence by sentence
We present a new discretization for advection-diffusion problems with Robin boundary conditions on complex time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme that uses a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the $L^1$, $L^2$, and $L^\infty$ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to handle conversion of one concentration field to another across a moving boundary.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present a new discretization for advection-diffusion problems with Robin boundary conditions on complex time-dependent domains. | contribution 1.00 | 0.93 | 0.19 | asserted 1.00 |
| s1 | The method is based on second order cut cell finite volume methods introduced by Bochkov et al. to discretize the Laplace operator and Robin boundary condition. | method 0.93 | 0.09 | 0.06 | asserted 1.00 |
| s2 | To overcome the small cell problem, we use a splitting scheme that uses a semi-Lagrangian method to treat advection. | method 0.96 | 0.87 | 0.06 | asserted 1.00 |
| s3 | We demonstrate second order accuracy in the $L^1$, $L^2$, and $L^\infty$ norms for both analytic test problems and numerical convergence studies. | result 1.00 | 0.96 | 0.37 | asserted 1.00 |
| s4 | We also demonstrate the ability of the scheme to handle conversion of one concentration field to another across a moving boundary. | result 0.91 | 0.94 | 0.14 | asserted 1.00 |
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Original abstract
We present a new discretization for advection-diffusion problems with Robin boundary conditions on complex time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme that uses a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the $L^1$, $L^2$, and $L^\infty$ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to handle conversion of one concentration field to another across a moving boundary.
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Recent works on finite volume cut-cell methods include the solution of Poisson's equation [ref] and the convection–diffusion equation [ref] with Robin boundary conditions.
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