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Finite difference methods for linear transport equations
Numerical Analysis · Submitted 21 Sep 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The key point for both of our methods is to obtain fine $L^2$-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna-Lions.
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The abstract, sentence by sentence
DiPerna-Lions (Invent. Math. 1989) established the existence and uniqueness results for linear transport equations with Sobolev velocity fields. This paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is $L^p$-strongly convergent. The second method is based on an implicit scheme with $L^2$-estimates, where the discrete Helmholtz-Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and $L^2$-strongly convergent. The key point for both of our methods is to obtain fine $L^2$-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna-Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method involving transport equations, where rigorous discrete approximation of geometric quantities of level sets is discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | DiPerna-Lions (Invent. | background 1.00 | 0.02 | 0.01 | unclear 0.97 |
| s1 | Math. 1989) established the existence and uniqueness results for linear transport equations with Sobolev velocity fields. | background 1.00 | 0.02 | 0.03 | asserted 1.00 |
| s2 | This paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. | contribution 1.00 | 0.78 | 0.11 | asserted 0.99 |
| s3 | The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is $L^p$-strongly convergent. | method 0.95 | 0.93 | 0.15 | asserted 1.00 |
| s4 | The second method is based on an implicit scheme with $L^2$-estimates, where the discrete Helmholtz-Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and $L^2$-strongly convergent. | method 0.93 | 0.93 | 0.10 | asserted 1.00 |
| s5 | The key point for both of our methods is to obtain fine $L^2$-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna-Lions. | contribution 0.35 | 0.86 | 0.49 | asserted 1.00 |
| s6 | Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method involving transport equations, where rigorous discrete approximation of geometric quantities of level sets is discussed. | contribution 0.76 | 0.88 | 0.06 | asserted 0.99 |
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Original abstract
DiPerna-Lions (Invent. Math. 1989) established the existence and uniqueness results for linear transport equations with Sobolev velocity fields. This paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is $L^p$-strongly convergent. The second method is based on an implicit scheme with $L^2$-estimates, where the discrete Helmholtz-Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and $L^2$-strongly convergent. The key point for both of our methods is to obtain fine $L^2$-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna-Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method involving transport equations, where rigorous discrete approximation of geometric quantities of level sets is discussed.
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1 arXiv paper cite this in 4 sentences
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We will follow this idea to deal with [ref], where local averaging of possibly unbounded velocity fields is used instead of the truncation used in [ref] in order to keep the discrete divergence-free constraint; an artificial boundary condition is imposed to the discretization of [ref]; the artificial boundary condition does not cause any harm to the solution, if the (locally averaged) velocity field vanishes on the boundary; since the support of the locally averaged velocity field can be slightly larger than $Ω$, [ref] will be solved on a domain larger than $Ω$ with constant-extension of the velocity field and the density field.
A finite difference method for inhomogeneous incompressible Navier-Stokes equations · Introduction · 0.81
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