Asymptotic safety in quantum gravity

In computational fluid dynamics QUICK, which stands for Quadratic Upstream Interpolation for Convective Kinetics, is a higher order differencing scheme that considers a three point upstream weighted quadratic interpolation for the cell phase values. In computational fluid dynamics there are many solution methods for solving the steady convection–diffusion equation. Some of the used methods are the central differencing scheme, upwind scheme, hybrid scheme, power law scheme and QUICK scheme.

The QUICK scheme was presented by Brian P. Leonard – together with the QUICKEST (QUICK with Estimated Streaming Terms) scheme – in a 1979 paper.

In order to find the cell face value a quadratic function passing through two bracketing or surrounding nodes and one node on the upstream side must be used. In central differencing scheme and second order upwind scheme the first order derivative is included and the second order derivative is ignored. These schemes are therefore considered second order accurate where as QUICK does take the second order derivative into account, but ignores the third order derivative hence this is considered third order accurate. This scheme is used to solve convection–diffusion equations using second order central difference for the diffusion term and for the convection term the scheme is third order accurate in space and first order accurate in time. QUICK is most appropriate for steady flow or quasi-steady highly convective elliptic flow.

Quadratic interpolation for QUICK scheme

Quadratic profile

For the one dimensional domain shown in the figure the Φ value at a control volume face is approximated using three-point quadratic function passing through the two bracketing or surrounding nodes and one other node on upstream side. In the figure, in order to calculatethe value of the property at the face, we should have three nodes i.e. two bracketing or surrounding nodes and one upstream node.

1. Φ_w when u_w > 0 and u_e > 0 a quadratic fit through WW, W and P is used,
2. Φ_e when u_w > 0 and u_e > 0 a quadratic fit through W, P and E is used,
3. Φ_w when u_w < 0 and u_e < 0 values of W, P and E are used,
4. Φ_e when u_w < 0 and u_e < 0 values of P, E and EE are used.

Let the two bracketing nodes be i and i − 1 and upstream node i – 2 then for a uniform grid the value of φ at the cell face between the three nodes is given by:

Φ_face = Template:Frac Φ_i-1 + Template:Frac Φ_i − Template:Frac Φ_i-2.

Interpretation of the property when the flow is in different directions

The steady convection and diffusion of a property ‘Ƥ’in a given one dimensional flow field with velocity ῾ u` and in the absence of sources is given

$\frac{d(\rho u\varphi )}{dx}=\frac{d}{dx}\left(r\frac{d\varphi }{dx}\right).$

For the continuity of the flow it must also satisfy

$\frac{d(\rho u)}{dx}=0.$

Discretizing the above equation to a control volume around a particular node we get

$(\rho uA\varphi {)}_e-(\rho uA\varphi {)}_w=(rA\frac{\partial \varphi }{\partial x}{)}_e-(rA\frac{\partial \varphi }{\partial x}{)}_w$

Integrating this continuity equation over the control volume we get

(ρuA)_e - (ρuA)_w = 0

now assuming F = ρu and D = r/δx

The corresponding cell face values of the above variables are given by

F_w = (ρu)_w

F_e = (ρu)_e

D_w = r_w /δx_WP

D_e = r_e /δx_PE

Assuming constant area over the entire control volume we get

F_eΦ_e – F_wΦ_w = D_e(Φ_E – Φ_P) – D_w (Φ_P - Φ_W)

Positive direction

When the flow is in positive direction the values of the velocities will be u_w > 0 and u_e > 0,

For “w (west face)” bracketing nodes are W and P, the upstream node is WW then,

Φ_w = 6/8Φ_W + 3/8Φ_P − 1/8Φ_WW

For “e (east face)” bracketing nodes are P and E, the upstream node is W then

Φ_e = 6/8Φ_P + 3/8Φ_E − 1/8Φ_W

Gradient of parabola is used to evaluate diffusion terms.

If F_w > 0 and F_e > 0 and if we use above equations for the convective terms and central differencing for the diffusion terms, the discretized form of the one-dimensional convection–diffusion transport equation will be written as:

F_eΦ_e – F_wΦ_w = D_e(Φ_E – Φ_P) – D_w (Φ_P - Φ_W)

F_e(6/8Φ_p + 3/8Φ_E – 1/8Φ_w) – F_W(6/8Φ_w + 3/8Φ_p – 1/8Φ_ww) = D_e(Φ_E – Φ_P) – D_W(Φ_p – Φ_w).

On re-arranging we get

[D_w – 3/8F_w + D_e + 6/8F_e] Φ_P = [D_w + 6/8F_w + 1/8F_e] Φ_W + [D_e – 3/8F_e] Φ_E – 1/8F_wΦ_WW,

now it can be written in the standard form:

a_PΦ_P = a_WΦ_W + a_EΦ_E + a_WWΦ_WW.

where:

aW	aE	aWW	aP
Dw + 6/8Fw + 1/8Fe	De - 3/8Fe	- 1/8Fw	aw + aE + aWW + (Fe - Fw)

Negative direction

When the flow is in negative direction the value of the velocities will be u_w < 0 and u_e < 0,

For west face w the bracketing nodes are P and E, upstream node is W and for the east face E the bracketing nodes are EE and E, upstream node is P

For F_w < 0 and F_e < 0 the flux across the west and east boundaries is given by the expressions :

Φ_w = 6/8Φ_P + 3/8Φ_W - 1/8Φ_E

Φ_e = 6/8Φ_E + 3/8Φ_P - 1/8Φ_EE

Substitution of these two formulae for the convective terms in the discretized convection-diffusion equation together with central differencing for the diffusion terms leads, after re-arrangement similar to positive direction as above, to the following coefficients.

aW	aE	aEE	aP
Dw + 3/8Fw	De - 6/8Fe - 1/8Fw	1/8Fe	aw + aE + aEE + (Fe - Fw)

QUICK scheme for 1-D convection–diffusion problems

a_PΦ_P = a_WΦ_W + a_EΦ_E + a_WWΦ_WW + a_EEΦ_EE

Here, a_P = a_W + a_E + a_WW + a_EE + (F_e - F_w)

other coefficients

aW	aWW	aE	aEE
Dw + 6/8 αw Fw + 1/8Fe αe +3/8 (1 – αw)Fw	-1/8 αwFw	De - 3/8αe Fe -6/8(1–αe)Fe -1/8 (1–αw)Fw	1/8(1 – αe)Fe

where

α_w=1 for F_w > 0 and α_e=1 for F_e > 0

α_w=0 for F_w < 0 and α_e=0 for F_e < 0.

comparing the solutions of QUICK and upwind schemes

From the below graph we can say that QUICK scheme solution is much more accurate than the upwind scheme because the solution of the QUICK scheme matches the exact solution with lot more accuracy than the upwind scheme. In this scheme we face the problems of undershoot and overshoot due to which some minor errors occur. so these overshoots and undershoots are to be considered while interpreting solutions. False diffusion errors will also get minimized when compared with other schemes.

Coparision of QUICK and UPWIND solutions

See also

• Upwind scheme
• Power law scheme
• Finite-volume method
• Finite difference method

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010