Jan Prokaj, Ph.D.
Founder

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Multirate numerical techniques for diffusion problems in one spatial dimension
Summary
Diffusion problems arise in many branches of science and engineering. The onedimensional diffusion equation is solved using a multirate numerical algorithm. The algorithm divides the system into different parts or blocks, allowing each block to take different time steps.
For each of the three initial profiles studied in tandem with three representative diffusion coefficient dependences, there is a very significant speedup over a CrankNicholson timestepping scheme without blocking. The greatest speedup (by a factor of 5 in certain cases) is obtained for a linear diffusion with relatively small diffusion coefficient. Diffusion speed and block size are found to be major factors affecting the performance of this algorithm. Implementation of this algorithm for a twodimensional diffusion is possible, and even greater speedups are expected in that case, since the second dimension allows for the inactive block(s) to occupy larger fractions of the overall domain.
Multirate timestepping
 Divides the diffusion system in multiple blocks

Each block has different timestepping rate
 Active blocks updated frequently
 Inactive blocks rarely
 Block boundaries updated as necessary
Implementation
 2 implementations
 Fixed blocking (updates inactive block at a fixed interval)
 Automatic blocking (updates inactive block whenever necessary)
 CrankNicholson numerical method used as a base
 Equation discretized using box method
 Diffusion model: D(C) = αC + β
 Neumann boundary conditions
 2 blocks
 Tridiagonal matrix solver
 Written in C language
What was tested

3 initial profiles:
 0.75*e^{0.5*(x1)2}
 3.50*e^{4.0*(x1)2}
 1.50*e^{2.0*(x1)2}
 Timestep size: 0.025, 0.05, 0.01
 Gridstep size: 0.025, 0.05, 0.01
 Nonlinear diffusion (α = 1.5, β = 0.02)

Linear diffusion
 α = 0, β = 0.02
 α = 0, β = 0.2

2 implementations of multirate algorithm
 Fixed blocking
 Automatic blocking
Results
Conclusions
 Multirate numerical methods produce significant speedup over singlerate methods

Magnitude of speedup depends on:
 Diffusion speed
 Size of the inactive block
 Timestep size (smaller > greater speedup)
 Gridstep size (greater > greater speedup)

Automatic blocking better than Fixed blocking
 No calculation necessary to determine the frequency of updating the inactive block in Automatic blocking
 But, no significant difference in speedup

Almost no loss of accuracy in the multirate algorithm
 Depends on the concentration value used to divide the system into blocks
Acknowledgements
This project was supported by a SMART grant.
Reference
J. Prokaj , S. Roy Choudhury. Multirate numerical techniques for diffusion problems in one spatial dimension. Internat. J. Appl. Sci. Comput , 13(3):126137, 2006.