need for interpolation schemes and flux limiters in FVM

AIM :- Need of interpolation schemes and flux limiters in FVM.

INTRODUCTION :- It is interested things that use of interpolation schemes and flux limiters in finite volume method (i.e. FVM).

INTERPOLATION METHODS :-

# The approximation of surface and volume integrals require values of the variable at locations other than the computational nodes of the CV.

# Values at these locations are obtained using interpolation formulae.

# Various possibilities for interpolation. e.g.

1. Upwind interpolation
2. Linear interpolation
3. Quadratic upwind interpolation
4. Hybrid, TVD and ENO interpolation schemes.

UPWIND INTERPOLATION (UDS):-

# UDS is equivalent to using a backward or forward finite difference approximation (depending on the flow direction).

ADVANTAGE OF UPWIND INTERPOLATIONS :-

Upwind interpolation (UDS) is the only approximation that satisfies boundedness criterion unconditionally.

DISADVANTAGE OF UPWIND INTERPOLATION :-

# UDS is numerically diffusive.

# Numerical diffusion is magnified in multidimensional problems if flow is oblique to the grid.

# Peaks or rapid variation in the variables will be smeared out.

# Only first order accurate. Hence, very fine grids are require to obtain accurate solutions.

# Approximate the values of the variable at CV face centre by linear interpolation of the values at two nearest computational nodes.

# The linear interpolation is equivalent to the use of central difference formula of the first order derivative and hence, this scheme is also difference scheme (CDS).

LINEAR INTERPOLATION CHARECTERASTICS :-

# CDS scheme is second order accurate

# It may produce oscillatory solutions.

# Simplest and most widely ised interpolation.

# Leads to a very simple approximation for evaluation of gradients require for evaluation of diffusive fluxes.

# Approximation of gradients has second order accuracy on uniform grids.

# Its formaly of first order on non-uniform grids.

FLUX LIMITER :-

Flux limiters are used inhigh resolution schemes i.e. numerical schemes used to solve problem in science and engineering, paricularly fluid dynamics, described by PDE\'s. They are used in high resolution schemes, such as the MUSCL scheme, to avoid the spurious oscillations(wiggles) that would otherwise occure with high order spatial descritization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution schemes, make the solution total variation diminishing (TVD).

Note that flux limiters are also reffered to as slope limiters because they both have the same mathmatical form, and both have the effect of limiting the solution gradient near shocks or discontinuities. In general, the term flux limiter is used when the limiter acts on system stetes (like pressure, velocity...etc).

FINITE VOLUME METHODS :-

The finite volume methods (FVM) is a method for representing and evaluating partial differential equation in the form of algebric equations. In the finite volume method, volume integral in a PDE\'s that cantain a divergence theorem. These term are then evaluated as fluxes at the surface of each finite volume. Because the flux entering a given volume is a identical to that leaving the adjecent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unconstructed meshes. The method is used in many CFD\'s packages. \"Finite Volume\" reffers to the small volume surrounding each node point on a mesh.

Alternative to the finite difference method or finite element method, values are calculated at descrete places on a meshed geometry.

Need for Interpolation schemes and flux limiters in FVM :-

• The Guass or divergence theorem simply states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
• This theorem is fundamental in the FVM, it is used to convert the volume integrals appearing in the governing equation into surface integrals.
• The face values appearing in the convective and diffusive fluxes have tobe computed by some form of interpolation from the centroid values of the control volume at both sides of face.
• After spacial discretization, we can proceed with the temporal discretization. By proceeding in this way we are using the method of lines (MOL).
• The main advantage of the MOL method, is that it allows us to select numerical approximations of different accuracy for the spacial and temporal terms. Each term can be treated differently to yield to different accuracies.
• Nearly all flows in nature and industrial applications are unsteady (also known as transient or time dependent).
• Unsteadiness is due to-
• Instabilities
• Non-equilibrium initial conditions
• Time dependent boundary conditions
• source term
• chemical reactions
• movieng or deforming bodies
• turbulence
• Buoyancy
• Convection
• Multiple phases

Steady Simulation vs Unsteady Simulations :-

• Steady simulation requre less computational power than unsteady simulations.
• They are also much fasters than unsteady simulations.
• But sometimes they do not converge to the right solution.
• They are easier to post process and analyse.
• We can use the solution of an unconverged steady simulation as initial conditions for an unsteady simulations.
• Remember, steady simulation are not time accurate, therefore we can not use them to compute temporal statistics or compute the shedding frequency.

Following are the plots for 1D linear convection equation are -

1D linear heat conduction equation :-

where, n = Grid points.

1. n=20, plot :-

In this plot where n=20, we can see that there is a drastic decrease in the velocity profile. At the uppar left corner somewhat gap formed which results at the it slowly decreases. The amplitude of the velocity profile keeps decreasing with the length, the minimum velocity is approximately 1.6 to 1.7 m/s.

2. n=40, plot :-

In this plot where n=40, we can see that there is a significant decrease in the velocity profile. In left uppar corner of the truncated part little gap compare to the previous graph which results at the end corner it drastic decreases. The amplitude of the velocity profile keeps decreasing with lengths. The minimum velocity is approximately between 1.7 to 1.8 m/s.

3. n=80, plots :-

In this plot where n=80, we can see that there is a gradual decrease in the velocity profile, the change in the velocity profile, the change with respect to length is not a major change. At the uppar left corner no gap formed, it means at the end smaller perpendicular angle formed. Hence, we can see that amplitude is almost constant The minimum and maximum velocity very close to 2 m/s.

4. n=160, plots :-

There is a disturbance in the velocity profile. We can see that there is a blow up and it noticed that ripple are formed in the plot making it unstable. Here is no any specific graph formed because is no constant velocity profile. For this value velocity profile gives unstable graph.

Comparison of the effect of Time step on the Numerical Solution :-

From the above graph, we can see that for the first time step the solution is blowing out implying that the solution is unstable. It can be explained with respect to the method used here i.e. an explicit method is very sensitive to time step given which results in an unstable solution. So, the time step has tobe correct or within the viable range for the solution. This can be determined by a condition called \'Courant-Friendrinches-Lewy (CFL) condition\'.It is necessary condition for convergence while solving certain PDE\'s (usualy hyperbolic PDE\'s) numerically. It arises in the numerical analysis of explicit time intigration schemes,. when these are used for the numerical solution. Accordingly the time steps taken for the simulation should be less than a certain value for a stable solutions.

CONCLUSION :-

In this way I desribe the need for interpolation schemes and flux limiters in FVM by explaining detailed terms. Also, I put 1D linear convection equation and describe each graph at every nodes and finded amplitude velocity, Also the different time step compare with each other and describe with detailed analysis.

Also describe steady state and unsteady state condition, and which one is better to compare with simulations.