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# Finite-Difference Schemes - Science topic

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Staggered grid finite-difference (FD) methods are widely used for elastic wave equation modelling because of their high computational efficiency, smaller memory requirement and easy implementation. I am looking for the mathematical foundation for higher order (8th) of finite difference staggered-grid method on 2D P-SV elastic wave propagation.

Dear all,

I am trying to couple a simulation with two codes, say code A and code B (different discretizations). For code A we solve simply the conduction equation (solid), while in B we solve the NS + temperature convection/diffusion (fluid).

The coupling is explicit, while the time scheme for each code is implicit.

at each iteration, code A gives code B the heat flux to impose at the boundary of domain B, while code B gives code A the temperature to impose at the boundary of domain A. So in short, the coupling is done via a Neumann/Dirichlet BCs.

I am facing stability issues that don't allow me to finish correctly the transient solution.

Let us focus only on domain A (code A). When code A receives the temperature values from code B to apply a Dirichlet BC, the values are stocked in a layer of cells; so called the ghost cells. We denote these values as T_wall.

Now, we need to compute the heat flux at the boundary to give to code B. There are two possibilities to compute such a flux :

1- use T_wall and one inner temperature value

2- use 2 or 3 inner temperature values to calculate the flux, and then extrapolate it to the boundary.

I am getting serious instabilities if I calculate the flux via the first approach (T_wall and 1 inner temperature value). Otherwise, with the second approach (only inner temperature values), the simulation is stable and a steady-state is reached.

Can I have please your opinion on this subject. Have you ever faced such kind of problems ?

Is it really true that computing the flux with the second approach leads to a stable solution, but to a non-consevative scheme ?

Best regards

Elie Saikali

I am solving a transinet convection-diffusion equation

u_t + a*u_x - nu*u_xx = 0 in (0,1)

Taking,

Convection coefficient a = 1

Diffusion coefficient nu = 0.005

Number of elements = 100

I got Peclet number Pe = 1

For same case now,

Time step = 0.01

End time = 0.6

I got Courant number C = 1

I need following 4 conditions to fulfill.

1). C =1 and Pe = 5

2). C =1 and Pe = 100

3). C =3 and Pe = 5

4). C =3 and Pe = 100

My question is:

What should be the value of

"Convection coefficient a",

"Number of elements",

"Diffusion coefficient nu",

"Time step" and

"End time"

to get these 4 results separately.

To solve the heat conduction (or diffusion) equation, it is well known that explicit schemes (like FTCS) are rarely used, because very tiny timesteps are required to avoid instability. I know that implicit methods usually have no such problem, but they have to handle huge matrices and cannot be trivially parallelized.

However, there are explicit methods with much better stability properties:

- Runge–Kutta–Chebyshev,

- ADE (Alternating Direction Explicit),

- Hopscotch,

- Dufort-Frankel (etc.?)

Now I can not really understand

**why**they haven't become widespread in the practice. Have they got serious disadvantages? If yes, what are them? Or maybe only because people are unaware of them?*I have the following problem:*

*u_t = u_xx + u_yy, in Omega, t > 0*

*u(x,y,0) = 1, in Omega*

*u(x,y,t) = 0, on dOmega, t > 0*

*when Omega is defined to be a domain between the rectangular 1x1 and the circle of radius 0.3 centered at (0.5,0.5). dOmega denotes the boundary of the domain Omega.*

I want to solve the problem

**numerically using explicit scheme**for dx = dy = 1/50. And I want to solve it in MATLAB.So far i got a general scheme for all the points, the problematic and non-problematic ones. But I don't know how to compute it.

I want some code that can, for each point, detect the value of alpha and beta (the coefficients in front of dx, dy when the distance to a neighboring grid point is less than dx or dy, i.e. when the neighboring grid point is on the boundary of the circle), for every point on the grid.

Can anyone help me figure out how to compute this?

I am solving a parabolic PDE with compact finite difference schemes and I don't have the exact solution for that. So how can I obtain the spatial order of convergence and temporal order of convergence separately for the above PDE because double mesh principle gives the global order of convergence (to the best of my knowledge).

I am trying to implement Maccormack finite-difference scheme (1969) to find the solution of HD (/MHD) equations in spherical polar coordinate. For simplicity axisymmetric condition can be assumed. The main problem is arising from the continuity equation : $\frac{\partial \rho}{\partial t} = \nabla (\rightoverarrow{v} \rho)$.

The divergence operator contains both the derivatives with respect to "r" and "\theta". I have some confusions how to implement the forward and backward difference with respect to "r" and "\theta" for predictor and corrector steps.

Is there any proper reference for this particular 2d/3d problem?

Anybody can explain why WENO scheme is not considered to be TVD.

Is there any implication of this fact on reconstruction data having large gradients ?

What about MPWENO, is this method abstain the emergence of oscillation in the vicinity of large gradients when using WENO?

Hi,

I want to implement WENO reconstruction on characteristic variables.

Therefore, the following transformation between primitve and characteristic variables has to be defined:

W= P

^{-1}Vwhere W is the characteristic variable vector,

V is the primitive variable vector

and P is the Jacobian matrix.

In the literature I saw it is common to write the above transformation as follows:

W

_{i}= (P_{i+1/2)}^{-1 }V_{i}P

_{i+1/2 }is to be evaluated at V_{i+1/2 }through arithmetic or Roe's average.For this example, we choose fifth-order spatial accurate reconstruction and therefore the three stencils are being used:

S

_{0}=(i,i+1,i+2), S_{1}=(i-1,i,i+1),S_{2}=(i-2,i-1,i)Then, one has to apply WENO reconstruction on Wi (characteristic vector) with the stencils indicated above and to get

W

_{L,i+1/2 }and W_{R,i-1/2}.I encountered some difficulties in doing so and I got oscillations in the vicinity of discontinuities.

- Why the projection matrix is "freezed" at V
_{i+1/2 }? Wht not just V_{i} - W
_{i }was calculated throught P_{i+1/2}, and then being used to reconstruct W_{R,i-1/2. }Isn't it wrong to use state variables data from the right neighnour in order to reconstruct the left side of a given computational cell? - In the matrix P there are terms involving the speed of sound c. How should this speed be calculated? The first option is to calculate the speed at each cell and average as: c
_{i+1/2}= 0.5(c_{i}+c_{i+1}). The second option is to calculate c at the cell face with the use of the primitive variables at the cell face,although then one has to average e.o.s parameters at the cell face as well. - Is there any simple numerical code (FORTRAN,C,MATLAB) that one can learn the use of characteristic WENO reconstruction ?

Thanks in advance for your replies.

ENO is a much better class of accurate scheme

Implementation is well reported in Shu-Osher Harten report from ICASE 1990

I want to calculate

**order of accuracy**of Finite Difference Scheme with**non-uniform grids**. Here**analytical solution**is not available. Please suggest some idea..I have problem with specifying blending factor 0.9 where can I specify this blending factor in FLUENT

Does anybody know how the derivatives of the coefficient functions in equation (4) of the paper "High-Order Difference Schemes for Two-Dimensional Elliptic Equations" by M. M. Gupta, R. P. Manohar and J. W. Stephenson are discretised? Assuming that the nine-point stencil should be kept, second-order central differences are a "natural" choice, I assume. But how can this lead to a fourth-order accurate scheme?

Thanks for your answers.

I try to solve numerically the differential equation with mixed derivative. I found one possibility for the mixed derivative like y(i+1,j-1)-y(i,j-1)-y(i-1,j+1)+y(i-1,j). But may be there are another ways...

Which dimensionless numbers should be taken into account in the stability analysis of a numerical method for Unsteady Convection-Diffusion Problems?

Which one shows more stability and accuracy of the numerical solutions?

1). Lax-Wendroff method with consistent mass matrix

2). Lax-Wendroff method with diagonal mass matrix

To approximate numerically the solution U we consider the Galerkin space discretization and time discretizations based on the Crank-Nicolson method and on the fourth-order Padé approximation R2,2.

What are the steps required to compute the approximate solution Un+1 at the time level tn+1 using both these discretization schemes?