Back and forth error compensation and correction methods for removing errors induced by uneven gradients of the level set function.

*(English)*Zbl 1031.65110The paper considers the reduction of numerical errors in the solution of time dependent linear advection equations which occur, e.g., in level set methods. Two schemes for error reduction are proposed. Both schemes, the back error compensation method as well as the forth error compensation method perform as a first step the solution of the equation with a basic scheme from time \(t_n\) to \(t_{n+1}\) and as second step a backward solution with the same scheme from \(t_{n+1}\) to \(t_{n}\).

Ideally, the solution on \(t_n\) should be recovered. However, due to numerical errors this is in general not the case. Based on the error, the backward error compensation error method defines as third step a modified solution in \(t_n\). With this modified solution, the final solution in \(t_{n+1}\) is computed. In the forward error compensation method, the third step is again a solve from \(t_n\) to \(t_{n+1}\) based on the solution obtained with the second step.

The csation of the error is now performed using the difference of the solution after the first and third step. For the case of applying the backward error compensation method to an ordinary differential equation, it is proven that this method improves the order of accuracy for certain schemes. In addition, a simple stability result is proven for the one-dimensional translation equation.

Numerical tests using as basic scheme an upwind scheme of first order are presented, in particular for Zalesak’s problem. They show a considerable improvement of accuracy using both error compensation approaches in comparison to using the basic scheme without error compensation. Finally, there is a short comparison of the presented approach to other methods of error reduction from the literature.

Ideally, the solution on \(t_n\) should be recovered. However, due to numerical errors this is in general not the case. Based on the error, the backward error compensation error method defines as third step a modified solution in \(t_n\). With this modified solution, the final solution in \(t_{n+1}\) is computed. In the forward error compensation method, the third step is again a solve from \(t_n\) to \(t_{n+1}\) based on the solution obtained with the second step.

The csation of the error is now performed using the difference of the solution after the first and third step. For the case of applying the backward error compensation method to an ordinary differential equation, it is proven that this method improves the order of accuracy for certain schemes. In addition, a simple stability result is proven for the one-dimensional translation equation.

Numerical tests using as basic scheme an upwind scheme of first order are presented, in particular for Zalesak’s problem. They show a considerable improvement of accuracy using both error compensation approaches in comparison to using the basic scheme without error compensation. Finally, there is a short comparison of the presented approach to other methods of error reduction from the literature.

Reviewer: Volker John (Magdeburg)

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35L45 | Initial value problems for first-order hyperbolic systems |

##### Keywords:

flux corrected transport; front tracking; level set method; numerical examples; comparison of methods; linear advection equations; backward error compensation method; error reduction; forward error compensation method; stability; translation equation; Zalesak’s problem##### Software:

SHASTA
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\textit{T. F. Dupont} and \textit{Y. Liu}, J. Comput. Phys. 190, No. 1, 311--324 (2003; Zbl 1031.65110)

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