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The estimation of truncation error by \(\tau \)-estimation revisited. (English) Zbl 1242.65222

Summary: The aim of this paper is to accurately estimate the local truncation error of partial differential equations, that are numerically solved using a finite difference or finite volume approach on structured and unstructured meshes. In this work, we approximate the local truncation error using the \(\tau \)-estimation procedure, which aims to compare the residuals on a sequence of grids with different spacing. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error for both finite difference and finite volume approaches on different grid topologies. Then, we extende the analysis to two-dimensional problems: first on linear and non-linear scalar equations and finally on the Euler equations. We demonstrate that this approach yields a highly accurate estimation of the truncation error if some conditions are fulfilled. These conditions are related to the accuracy of the restriction operators, the choice of the boundary conditions, the distortion of the grids and the magnitude of the iteration error.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65L70 Error bounds for numerical methods for ordinary differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N08 Finite volume methods for boundary value problems involving PDEs
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
35J60 Nonlinear elliptic equations
65L12 Finite difference and finite volume methods for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Software:

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References:

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