# zbMATH — the first resource for mathematics

Back and forth error compensation and correction methods for removing errors induced by uneven gradients of the level set function. (English) Zbl 1031.65110
The 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.

##### 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
SHASTA
Full Text:
##### References:
  Adalsteinsson, D.; Sethian, J.A., A fast level set method for propagating interfaces, J. comput. phys., 118, 269-277, (1993) · Zbl 0823.65137  Boris, J.P.; Book, D.L., Flux-corrected transport i. shasta, a fluid transport algorithm that works, J. comput. phys., 11, 38-69, (1973) · Zbl 0251.76004  Enright, D.; Fedkiw, R.; Ferziger, J.; Mitchell, I., A hybrid particle level set method for improved interface capturing, J. comput. phys., 183, 83-116, (2002) · Zbl 1021.76044  Fedkiw, R.P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 457-492, (1999) · Zbl 0957.76052  J. Gomes, O. Faugeras, Advancing and keeping a signed distance function, Technical Report 3666, INRIA, 1999  Harten, A.; Osher, S.; Engquist, B.; Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes, III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067  Liu, X.D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200-212, (1994) · Zbl 0811.65076  R.W. MacCormack, AIAA Paper 69-354, 1969  Merriman, B.; Bence, J.; Osher, S., Motion of multiple functions: a level set approach, J. comput. phys., 112, 334, (1994)  D. Nguyen, F. Gibou, R. Fedkiw, A fully conservative ghost fluid method and stiff detonation waves, 12th International Detonation Symposium, San Diego, CA, 2002  Osher, S.; Sethian, J., Fronts propagating with curvature-dependent speed: algorithms based on hamilton – jacobi equations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132  Prada, C.; Wu, F.; Fink, M., The iterative time reversal mirror: a solution to self-focusing in the pulse echo mode, J. acoust. soc. am., 90, 1119-1129, (1991)  J. Qiu, C. Shu, Finite difference WENO schemes with Lax-Wendroff type time discretization, SIAM J. Sci. Comput., to appear · Zbl 1034.65073  Sussman, M.; Fatemi, E., An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. sci. comput., 20, 1165-1191, (1999) · Zbl 0958.76070  Sussman, M.; Puckett, E.G., A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows, J. comput. phys., 162, 301-337, (2000) · Zbl 0977.76071  Sussman, M.; Smereka, P.; Osher, S., A level set method for computing solutions to incompressible two-phase flow, J. comput. phys., 119, 146-159, (1994) · Zbl 0808.76077  Zalesak, S.T., Fully multidimensional flux-corrected transport, J. comput. phys., 31, 335-362, (1979) · Zbl 0416.76002  Zhao, H.-K.; Chan, T.; Merriman, B.; Osher, S., A variational level set approach to multiphase motion, J. comput. phys., 127, 179-195, (1996) · Zbl 0860.65050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.