Stability and Accuracy of Diffusion Equation Solution

In summary: If the simulationproduced a better fit to my experimental data than the analyticsolution, then I would use the numerical solution in my design.Yes, this is probably the only thing you can do. I've used it in heat transferstudies to verify/validate finite element simulations compared to analyticheat diffusion solutions for regular geometric shapes. Eg, if I wasmodeling the cooling of an avocado, I would test the numericalsimulation against spheres and short columns. If the simulationproduced a better fit to my experimental data than the analyticsolution, then I would use the numerical solution in my design.
  • #1
baseball07
5
0
I have a general question about the solution to the Diffusion equation using the explicit finite difference method. Now, it is known the solution is stable when D*dt/dx^2 is less than 0.5, based on the choice of time and space steps. However, how does the choice of the time and space steps affect the actual numerical solution values, say if we were to compare with some experimental data? That is, a D*dt/dx^2 of 0.48 and 0.3 are both indeed stable, but they will both have different numerical values, correct? So how does one choose the correct time and space steps to get the closest solution?
 
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  • #2
baseball07 said:
I have a general question about the solution to the Diffusion equation using the explicit finite difference method. Now, it is known the solution is stable when D*dt/dx^2 is less than 0.5, based on the choice of time and space steps. However, how does the choice of the time and space steps affect the actual numerical solution values, say if we were to compare with some experimental data? That is, a D*dt/dx^2 of 0.48 and 0.3 are both indeed stable, but they will both have different numerical values, correct? So how does one choose the correct time and space steps to get the closest solution?

You can compare against analytic solutions. You can set the value of D*dt/dx^2 constant, and solve the equations using progressively smaller values of dx and dt (at fixed D*dt/dx^2 ) until you are satisfied that the solutions have stopped changing (to your satisfaction). You can also try this for several values of D*dt/dx^2 and compare results.
 
  • #3
Chestermiller said:
You can compare against analytic solutions. You can set the value of D*dt/dx^2 constant, and solve the equations using progressively smaller values of dx and dt (at fixed D*dt/dx^2 ) until you are satisfied that the solutions have stopped changing (to your satisfaction). You can also try this for several values of D*dt/dx^2 and compare results.

Yes, this is probably the only thing you can do. I've used it in heat transfer
studies to verify/validate finite element simulations compared to analytic
heat diffusion solutions for regular geometric shapes. Eg, if I was
modeling the cooling of an avocado, I would test the numerical
simulation against spheres and short columns.
 
Last edited:

Related to Stability and Accuracy of Diffusion Equation Solution

1. What is the diffusion equation and why is it important?

The diffusion equation is a partial differential equation that describes the behavior of a diffusing substance over time. It is important because it is used to model a wide variety of phenomena, including heat transfer, mass transfer, and chemical reactions, in fields such as physics, chemistry, biology, and engineering.

2. What factors affect the stability and accuracy of a diffusion equation solution?

The stability and accuracy of a diffusion equation solution are affected by various factors, including the choice of numerical method, the time and spatial discretization, the initial and boundary conditions, and the properties of the diffusing substance. In general, a more accurate and stable solution can be achieved by using a finer mesh, a smaller time step, and a more appropriate numerical method for the specific problem.

3. How can we assess the stability and accuracy of a diffusion equation solution?

Stability can be assessed by analyzing the behavior of the solution over time and ensuring that it does not grow or decay uncontrollably. Accuracy can be assessed by comparing the numerical solution to analytical solutions or experimental data, and by examining the convergence of the solution as the mesh and time step are refined.

4. What are some common challenges in solving the diffusion equation numerically?

Some common challenges in solving the diffusion equation numerically include dealing with complex geometries, handling nonlinear or time-dependent diffusion coefficients, and incorporating additional physical effects such as convection and reaction. These challenges often require the use of advanced numerical techniques and careful consideration of the overall solution strategy.

5. How can we improve the stability and accuracy of a diffusion equation solution?

To improve the stability and accuracy of a diffusion equation solution, it is important to carefully select and validate the numerical method, choose appropriate time and spatial discretizations, and properly account for any additional physical effects. It is also helpful to perform sensitivity analyses and grid convergence studies to identify any potential sources of error and ensure that the solution is reliable.

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