Analysis of Solution Stability in a 2D Heat conduction problem

ANALYSIS OF NUMERICAL STABILITY OF VARIOUS ITERATIVE SOLVERS FOR TRANSIENT 2D HEAT CONDUCTION:

[Part: 3/3]

INTRODUCTION:

The criterion of stability of a numerical scheme is determined by the way the errors propagate while the solution moves from one time-step to the next in case of a transient solver. While solving governing equations in the PDE form the discretized equations have error terms which should theoretically be minimized or should oscillate with a damped amplitude minimizing gradually as one moves forward in time, we define this important criterion as the CFL Number which stands for the Courant-Lewy-Friedrichs Condition which is essential for convergence of a solution by the method of finite differences. It also sets the condition that if the time-step of the problem is larger than a specific value the solution of the finite difference equation becomes impossible by some solvers.

The condition for the 2 Dimensional case is given below:

`C = alpha*(1/(Deltax)^2 + 1/(Deltay)^2)*Deltat`

where:
`alpha` = Thermal Diffusivity (Units = `m^2s^-1`)
`Deltax = Deltay` = Number of grid points in the x or y direction (Units = m)
`Deltat` = Time-step

Where `C <= C_(max) = 0.25` for convergence. Here `C_max = 0.25` is for the explicit solvers.

Implicit Solvers are un-conditionally stable and are less sensitive to numerical instability so they can tolerate a much larger value of `C` where an explicit solver might fail.

I. Explicit Solution Scheme:

For a time-step of dt = 0.01 s; CFL Number is 0.0025 which is way below `C_(max)` henc the solution shows good convergence:

dt 0.001

For a time-step of dt = 0.1 s; the CFL Number is 0.025 which is still below `C_(max)` indicating good convergence. However the solution resembles the steady state due to the number of iterations performed remains the same so the solution marches more farther in time than before.

 dt 0.1 cfl 0.025

At this point if we set time-step dt = 1 to obtain CFL Number = 0.25, the solution still remains stable as shown below:

dt 1 cfl 0.25

From this point any slight increase in time-step would render the solution unstable:

Temperature profile for time-step dt = 1.01; CFL Number = 0.2525

dt 1.01 cfl 0.2525

The above solution is unstable and highly divergent.

Temperature profile for time-step dt = 1.02; CFL Number = 0.2550

dt 1.02 cfl 0.2550

The solution is unstable and highly divergent.

Temperature profile for dt = 1.03; CFL Number = 0.2575:

cfl 1.03 cfl 0.2575

At this point the solution is too unstable and very highly divergent. Any further increase in dt would render the solver incapable of plotting the solution altogether.

dt = 1.04; CFL Number = 0.2600

dt 1.04 cfl 2600

dt = 1.05; CFL Number = 0.2625:

dt 1.05 cfl 0.2625

 II. Implicit Solution Scheme:

Implicit Iterative Schemes (Gauss Siedel, Gauss-Jacobi and Successive Relaxation) solvers used for the above problem are unconditionally stable and do not show any instability even at CFL Number = 1 The results are shown below:

A. Gauss Jacobi Method:

jac imp 1Gauss Siedel Method:

gs cfl 1

C. Successive Over Relaxation (SOR) Method:

sor imp cfl 1

The results resemble the steady state as expected for such a long period of heat conduction within the plate. The results do not deviate amongst the solver type used verifying that the implicit solvers are not suceptible to numerical instability.

Other Results obtained for time-step dt = 1 and CFL Number = 0.25 are shown below:

cfl 1 imp jac

 REMARKS:

Grid Spacing for the problem: `Deltax = Deltay = 0.02`

Thermal Diffusivity for the problem: `alpha = 1e-4`


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