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<p>Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by <i>h</i> and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Delta t\)</EquationSource> </InlineEquation> respectively, we prove an error estimate <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O(\Delta t^3 + \frac{h^4}{\Delta t})\)</EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-norm theoretically, which justifies the above-mentioned prediction if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(h = O(\Delta t)\)</EquationSource> </InlineEquation>. We also derive error bounds in the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H^1\)</EquationSource> </InlineEquation>- and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(H^2\)</EquationSource> </InlineEquation>-norms. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation> projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation.</p>

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Error estimates of the cubic interpolated pseudo-particle scheme for one-dimensional advection equations

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  • 发表日期：

    2026年2月2日

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  • Takahito Kashiwabara,
  • Haruki Takemura

摘要

Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by h and \(\Delta t\) respectively, we prove an error estimate \(O(\Delta t^3 + \frac{h^4}{\Delta t})\) in \(L^2\) -norm theoretically, which justifies the above-mentioned prediction if \(h = O(\Delta t)\) . We also derive error bounds in the \(H^1\) - and \(H^2\) -norms. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the \(L^2\) projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation.