<p>This article establishes the usefulness of the Smoothness-Increasing Accuracy-Conserving (SIAC) filter for reducing the errors in the mean and variance for a wave equation with uncertain coefficients solved via generalized polynomial chaos (gPC) whose coefficients are approximated using discontinuous Galerkin (DG-gPC). Theoretical error estimates that utilize information in the negative-order norm are established. While the gPC approximation leads to order of accuracy of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$m-1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> for a sufficiently smooth solution (smoothness of <i>m</i> in random space), the approximated coefficients solved via DG improves from order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2k+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for a solution of smoothness <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2k+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> in physical space. Our numerical examples verify the performance of the filter for improving the quality of the approximation and reducing the numerical error and significantly eliminating the noise from the spatial approximation of the mean and variance. Further, we illustrate how the errors are affected by both the choice of smoothness of the kernel and number of function translates in the kernel. Hence, this article opens the applicability of SIAC filters to other hyperbolic problems with uncertainty, and other stochastic equations.</p>

SIAC Accuracy Enhancement of Stochastic Galerkin Solutions for Wave Equations with Uncertain Coefficients

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发表日期：

2026年3月24日

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Andrés Galindo-Olarte,
Jennifer K. Ryan

摘要

This article establishes the usefulness of the Smoothness-Increasing Accuracy-Conserving (SIAC) filter for reducing the errors in the mean and variance for a wave equation with uncertain coefficients solved via generalized polynomial chaos (gPC) whose coefficients are approximated using discontinuous Galerkin (DG-gPC). Theoretical error estimates that utilize information in the negative-order norm are established. While the gPC approximation leads to order of accuracy of $m-1/2$ $m - 1 / 2$ for a sufficiently smooth solution (smoothness of m in random space), the approximated coefficients solved via DG improves from order $k+1$ $k + 1$ to $2k+1$ $2 k + 1$ for a solution of smoothness $2k+2$ $2 k + 2$ in physical space. Our numerical examples verify the performance of the filter for improving the quality of the approximation and reducing the numerical error and significantly eliminating the noise from the spatial approximation of the mean and variance. Further, we illustrate how the errors are affected by both the choice of smoothness of the kernel and number of function translates in the kernel. Hence, this article opens the applicability of SIAC filters to other hyperbolic problems with uncertainty, and other stochastic equations.