1,015 research outputs found
Flux Splitting for stiff equations: A notion on stability
For low Mach number flows, there is a strong recent interest in the
development and analysis of IMEX (implicit/explicit) schemes, which rely on a
splitting of the convective flux into stiff and nonstiff parts. A key
ingredient of the analysis is the so-called Asymptotic Preserving (AP)
property, which guarantees uniform consistency and stability as the Mach number
goes to zero. While many authors have focussed on asymptotic consistency, we
study asymptotic stability in this paper: does an IMEX scheme allow for a CFL
number which is independent of the Mach number? We derive a stability criterion
for a general linear hyperbolic system. In the decisive eigenvalue analysis,
the advective term, the upwind diffusion and a quadratic term stemming from the
truncation in time all interact in a subtle way. As an application, we show
that a new class of splittings based on characteristic decomposition, for which
the commutator vanishes, avoids the deterioration of the time step which has
sometimes been observed in the literature
A note on adjoint error estimation for one-dimensional stationary balance laws with shocks
We consider one-dimensional steady-state balance laws with discontinuous
solutions. Giles and Pierce realized that a shock leads to a new term in the
adjoint error representation for target functionals.This term disappears if and
only if the adjoint solution satisfies an internal boundary condition.
Curiously, most computer codes implementing adjoint error estimation ignore the
new term in the functional, as well as the internal adjoint boundary condition.
The purpose of this note is to justify this omission as follows: if one
represents the exact forward and adjoint solutions as vanishing viscosity
limits of the corresponding viscous problems, then the internal boundary
condition is naturally satisfied in the limit
ΠΠ½Π²Π΅ΡΡΠΈΡΠΈΠΎΠ½Π½Π°Ρ ΠΏΡΠΈΠ²Π»Π΅ΠΊΠ°ΡΠ΅Π»ΡΠ½ΠΎΡΡΡ ΠΏΡΠ΅Π΄ΠΏΡΠΈΡΡΠΈΠΉ-ΡΠ΅Π·ΠΈΠ΄Π΅Π½ΡΠΎΠ² ΠΎΡΠΎΠ±ΡΡ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΠΈΡ Π·ΠΎΠ½ Π½Π° ΠΏΡΠΈΠΌΠ΅ΡΠ΅ Π’ΠΎΠΌΡΠΊΠΎΠΉ ΠΎΠ±Π»Π°ΡΡΠΈ
Π ΡΡΠ°ΡΡΠ΅ ΠΎΠ±ΠΎΠ±ΡΠ°ΡΡΡΡ ΠΈ Π²ΡΡΠ²Π»ΡΡΡΡΡ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΠΈ ΠΏΠΎΠ²ΡΡΠ΅Π½ΠΈΡ ΠΈΠ½Π²Π΅ΡΡΠΈΡΠΈΠΎΠ½Π½ΠΎΠΉ ΠΏΡΠΈΠ²Π»Π΅ΠΊΠ°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΠΏΡΠ΅Π΄ΠΏΡΠΈΡΡΠΈΠΉ ΠΏΡΠΈ Π²Ρ
ΠΎΠΆΠ΄Π΅Π½ΠΈΠΈ Π² ΠΎΡΠΎΠ±ΡΡ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΡΡ Π·ΠΎΠ½Ρ (ΠΠΠ), Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΎΠΏΡΠΎΡΠ° ΠΈΡΡΠ»Π΅Π΄ΡΠ΅ΡΡΡ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠ΅ ΠΈΠ½ΠΎΡΡΡΠ°Π½Π½ΡΡ
ΠΏΠ°ΡΡΠ½Π΅ΡΠΎΠ² ΠΊ ΠΏΡΠΈΠ²Π»Π΅ΠΊΠ°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΡΠ΅Π·ΠΈΠ΄Π΅Π½ΡΠΎΠ² ΠΠΠ Π΄Π»Ρ ΠΈΠ½Π²Π΅ΡΡΠΈΡΠΈΠΉ ΠΈ Π΄ΠΎΠ»Π³ΠΎΡΡΠΎΡΠ½ΠΎΠ³ΠΎ ΡΠΎΡΡΡΠ΄Π½ΠΈΡΠ΅ΡΡΠ²Π°, Π°Π½Π°Π»ΠΈΠ·ΠΈΡΡΡΡΡΡ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΡ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠΎΠ² ΡΠ°Π±ΠΎΡΡ ΠΊΠΎΠΌΠΏΠ°Π½ΠΈΠΉ ΠΏΠΎΡΠ»Π΅ Π²Ρ
ΠΎΠΆΠ΄Π΅Π½ΠΈΡ Π² ΠΠΠ Π½Π° ΠΏΡΠΈΠΌΠ΅ΡΠ΅ ΠΠΠ Π’ΠΎΠΌΡΠΊΠΎΠΉ ΠΎΠ±Π»Π°ΡΡΠΈ
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