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Published on: June 2, 2010
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General spline filters for discontinuous Galerkin solutions
1Dept C.I.S.E., CSE Bldg, University of Florida, Gainesville, FL 32611-6120, United States.
Computers & Mathematics with Applications (Oxford, England : 1987)
|November 24, 2015
Summary
The discontinuous Galerkin (dG) method uses Smoothness-Increasing Accuracy-Conserving (SIAC) convolution for enhanced accuracy. This study provides simple formulas for optimal SIAC spline coefficients, even with non-uniform knots.
Area of Science:
- Numerical analysis
- Computational mathematics
Background:
- The discontinuous Galerkin (dG) method generates polynomial sequences.
- Post-processing dG outputs with Smoothness-Increasing Accuracy-Conserving (SIAC) convolution can improve accuracy and smoothness.
- Optimal SIAC convolution requires kernels that reproduce polynomials of degree 2d, where d is the B-spline degree.
Purpose of the Study:
- To derive simple formulas for computing optimal SIAC spline coefficients.
- To address the general case, including non-uniform knots.
Main Methods:
- Derivation of formulas for optimal SIAC spline coefficients.
- Utilizing B-splines of degree d to construct SIAC kernels.
Main Results:
- Simple formulas for optimal SIAC spline coefficients are presented.
- The formulas are applicable to general cases, including non-uniform knots.
- The derived coefficients ensure SIAC kernels reproduce polynomials of degree 2d.
Conclusions:
- The derived formulas simplify the computation of optimal SIAC spline coefficients.
- This advancement enhances the accuracy and superconvergence properties of dG methods.
- The findings are valuable for applications requiring high-accuracy numerical solutions.
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