Enriched Finite Volume Approximations of the Plane-Parallel Flow at a Small Viscosity

Gie, Gung-Min; Jung, Chang-Yeol; Lee, Hoyeon

doi:10.1007/s10915-020-01259-0

Cited by 4 publications

(3 citation statements)

References 18 publications

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“…Note that our analysis and computations on (1) generalize significant amount the earlier work [10] as it includes the case when

M

is singular and/or not diagonalizable. Using the techniques in singular perturbation analysis; see, for example, [11–18] as well as [19–28], we decompose the solution

{bold-italicu}^{ϵ}

of (1) as the sum of a slow (non‐stiff) part and fast (stiff) part, and construct an explicit and smooth function, called, corrector which approximates the fast (stiff) part of

{bold-italicu}^{ϵ}

. By embedding the explicitly built correctors in the conventional IF and ETD methods, and then apply the Runge–Kutta of order 2 and 4, we construct our semi‐analytic enriched IF and ETD schemes with Runge–Kutta methods.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, to achieve a sufficiently accurate approximation of the solution near the boundary, a massive mesh refinement is usually required, near the boundary, for the most classical numerical schemes. Instead of introducing massive mesh refinements, new semi-analytic methods have been proposed, see, e.g., [6,8,9,12,15,16]. The main component of this semi-analytic method is enriching the basis of traditional numerical methods, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In our algorithm, we split our test function into two parts, slow and fast gradient components, and learn effectively the two components simultaneously. This approach is motivated from an enriched basis method in numerical PDEs [8,9,15,16].…”

Section: Introductionmentioning

confidence: 99%

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Semi‐analytic time differencing methods for singularly perturbed initial value problems

Gie

Jung

Lee

2021

Numerical Methods Partial

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We implement our new semi‐analytic time differencing methods, applied to singularly perturbed non‐linear initial value problems. It is well‐known that, concerning the singularly perturbed initial problem, a very stiff layer, called initial layer, appears when the perturbation parameter is small, and this stiff initial layer causes significant difficulties to approximate the solution. To improve numerical quality of the classical integrating factor (IF) methods and exponential time differencing (ETD) methods for stiff problems, we first derived the so‐called correctors, which are analytic approximations of the stiff part of the solution. Then, by embedding these correctors into the IF and ETD methods, we build our new enriched schemes to improve the IF Runge–Kutta and ETD Runge–Kutta schemes. By performing numerical simulations, we verify that our new enriched schemes give much better approximations of solutions to the stiff problems, compared with the classical schemes without using the correctors.

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“…Note that our analysis and computations on (1) generalize significant amount the earlier work [10] as it includes the case when

M

is singular and/or not diagonalizable. Using the techniques in singular perturbation analysis; see, for example, [11–18] as well as [19–28], we decompose the solution

{bold-italicu}^{ϵ}

of (1) as the sum of a slow (non‐stiff) part and fast (stiff) part, and construct an explicit and smooth function, called, corrector which approximates the fast (stiff) part of