After Mid-Sem Flashcards
Q: Numerical Diffusion/ Arificial Viscosity/ Numerical Viscoisity?
Week 8 - Mid-Sem Week
Week 9 -
Q: Numerical Diffusion/ Arificial Viscosity/ Numerical Viscoisity?
Finite difference discretizations introduce some type of numerical error which results in numerical diffusion/ dispersion.
If the leading order term is proportional to even-order spatial derivative - diffusion, odd-order spatial derivative - dispersion.
A certain amount of diffusion is necessary to ensure numerical stability & dampens small scale irregularities.
Q: FTCS unstable? Lax-Modification?
Q: FTCS unstable? Lax-Modification?
Lax modification results in stabilization of the FTCS scheme by introducing numerical diffusion.
Q: Lax-Wendroff Method?
Q: Lax-Wendroff Method?
2nd order accurate both in space and time
first order upwinds (FTFS,FTBS) offer numerical stability but are low order accurate.
Q: What is Dispersion & Diffusion? When is Dispersion or Diffusion?
Q: What is Dispersion & Diffusion? When is Dispersion or Diffusion?
Leading order error term if odd-order derivative - dispersion
Leading order error term if even-order derivative - diffusion
Diffusion - amplitude gets reduced
Dispersion - smaller waves with varying wavelength travelling at different speed
Q: Leap-frog method?
Q: Leap-frog method?
CTCS scheme & it might lead to split solutions, neutrally stable
Q: Implicit methods?
Q: Implicit methods?
unconditionally stable like BTCS, CN
Q: Why Modified Wave Number Analysis?
Week 10
Lect-23
Modified Wave Number & Modified wave speed.
Q: Why Modified Wave Number Analysis?
The modified wave number analysis is used to evaluate a numerical method and properties like dissipation, dispersion can be determined.
Q: Simplest mathematical form of wave?
Q: Simplest mathematical form of wave?
expression for phi, d phi, d^2 phi
Q: Quality of FD scheme? about ik*
Q: Quality of FD scheme? about ik*
closer the modified wave number to the actual wave number, better the accuracy of scheme.
ik* can be real or complex, its form depends on choice of FD scheme.
its real part signifies the amplitude (numerical derivative) of the wave
its imaginary part corresponds to the phase shift from the exact derivative.
Q: Plot of modified wave number
Q: Plot of modified wave number
higher order schemes have numerical derivative close to the exact derivative for large wavenumbers.
if a higher-order scheme is used then small scales are accurately resolved (high wavenumbers means smaller wavelengths (small scale motions)) but higher-order schemes are computationally expensive also.
Q: Modified wave number analysis of linear advection equation using CD & upwind scheme?
Q: Modified wave number analysis of linear advection equation using CD & upwind scheme?
for CD: the each wave slows down
for upwind scheme: results in both phase and amplitude error
Q: Transportive property?
Q: Transportive property?
perturbation is convected only in the direction of velocity - transportive property of scheme
Q: Transportive property of FTCS & FTFS for linear advection equation?
Q: Transportive property of FTCS & FTFS for linear advection equation?
Check the perturbation at j-1, j & j+1 at (n+1)th time level given perturbation at j at nth time level
FTCS doesn’t respect the transportive property [& we know FTCS is unconditionally unstable]
Upwind scheme FTFS respects this property
Q: What is Conservative form & Non-conservative form? Why?
Q: What is Conservative form & Non-conservative form? Why?
if you add up the discretization terms over a grid, only the boundary terms should remain & the intermediate points should cancel out
upwind schemes are conservative in nature.
