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A POSTERIORI ERROR ANALYSIS OF TWO STAGE COMPUTATION METHODS WITH APPLICATION TO EFFICIENT DISCRETIZATION AND THE
Jehanzeb Hameed Chaudhry1, Don Estep2, Simon Tavener3
1Department of Mathematics & Statistics, The University of New Mexico, Albuquerque, NM 87131.
SIAM Journal on Numerical Analysis
|October 31, 2017
Summary
This study introduces a two-stage numerical method for solving initial value problems. The approach enhances accuracy and efficiency by combining coarse and fine discretization solutions, improving error estimation and parallel computing.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Scientific Computing
Background:
- Initial value problems (IVPs) are fundamental in science and engineering.
- Existing numerical methods face challenges in accuracy, efficiency, and error control.
- Two-stage computational approaches offer potential for improved performance.
Purpose of the Study:
- To develop a general framework for two-stage numerical methods for IVPs.
- To introduce a robust a posteriori error analysis for these methods.
- To enhance the efficiency and accuracy of solving complex computational problems.
Main Methods:
- Formulation of a general two-stage computation strategy.
- Development of a posteriori error analysis using computable residuals and adjoint problems.
- Application to dual-weighted error estimation and the Parareal Algorithm.
Main Results:
- A generalized error analysis accommodating variations in two-stage computations and adjoint problem formulations.
- Computation of dual-weighted a posteriori error estimates.
- Development of novel algorithms for efficient solutions considering error cancellation.
Conclusions:
- The proposed two-stage approach provides a flexible and effective framework for numerical solutions of IVPs.
- The a posteriori error analysis enables accurate error estimation and algorithm development.
- The methods show promise for improving efficiency in parallel-in-time schemes and adjoint problem solutions.
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