Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A space–time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quadtree data structures. For adaptive time integration, an embedded Runge–Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compression with respect to the uniform grid discretization are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Aftosmis, M.J.: Solution adaptive Cartesian grid methods for aerodynamic flows with complex geometries. von Karman Institute for Fluid Dynamics Lecture Series 1997–02, Rhode-Saint-Genèse (1997)

  2. Appelö, D., Hagstrom, T.: On advection by Hermite methods. Pac. J. Appl. Math. 4(2), 125–139 (2011)

    Google Scholar 

  3. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82(1), 64–84 (1989)

    Article  MATH  Google Scholar 

  5. Berger, M., LeVeque, R.: Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35(6), 2298–2316 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53(3), 484–512 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brun, E., Guittet, A., Gibou, F.: A local level-set method using a hash table data structure. J. Comput. Phys. 231(6), 2528–2536 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chiavassa, G., Donat, R.: Point value multiscale algorithms for 2D compressible flows. SIAM J. Sci. Comput. 23(3), 805–823 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chidyagwal, P., Nave, J.C., Rosales, R., Seibold, B.: A comparative study of the efficiency of jet schemes. Int. J. Numer. Anal. Model. Ser. B 3(3), 297–306 (2012)

    MathSciNet  Google Scholar 

  10. Cohen, A.: Wavelet methods in numerical analysis. In: Ciarlet, P., Lions, J. (eds.) Solution of Equation in \({\mathbb{R}}^n\) (Part 3), Techniques of Scientific Computing (Part 3), Handbook of Numerical Analysis, vol. 7, pp. 417–711. Elsevier, Amsterdam (2000)

    Google Scholar 

  11. Dahmen, W., Gotzen, T., Melian, S., Müller, S.: Numerical simulation of cooling gas injection using adaptive multiresolution techniques. Comput. Fluids 71, 65–82 (2013)

    Article  Google Scholar 

  12. Deiterding, R.: Block-structured adaptive mesh refinement: theory, implementation and application. ESAIM Proc. 34, 97–150 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. DeVore, R.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)

    Article  MathSciNet  Google Scholar 

  14. Domingues, M., Gomes, S., Roussel, O., Schneider, K.: An adaptive multiresolution scheme with local time stepping for evolutionary PDEs. J. Comput. Phys. 227(8), 3758–3780 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Domingues, M., Gomes, S., Roussel, O., Schneider, K.: Adaptive multiresolution methods. ESAIM Proc. 34, 1–96 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dormand, J.R., Prince, P.J.: New Runge–Kutta algorithms for numerical simulation in dynamical astronomy. Celest. Mech. 18, 223–232 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Harten, A.: Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Commun. Pure Appl. Math. 48(12), 1305–1342 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harten, A.: Multiresolution representation of data: a general framework. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kaibara, M., Gomes, S.: A fully adaptive multiresolution scheme for shock computations. In: Toro, E. (ed.) Godunov Methods: Theory and Applications, pp. 503–597. Kluwer/Plenum, Dordrecht/New York (2001)

    Google Scholar 

  20. Lindsay, K., Krasny, R.: A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow. J. Comput. Phys. 172(2), 879–907 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. ACM Trans. Graph. TOG 23(3), 457–462 (2004)

    Article  Google Scholar 

  23. Martin, D.F., Colella, P.: A cell-centered adaptive projection method for the incompressible Euler equations. J. Comput. Phys. 163(2), 271–312 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Min, C., Gibou, F.: A second order accurate level set method on non-graded adaptive Cartesian grids. J. Comput. Phys. 225(1), 300–321 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Müller, S.: Adaptive multiscale schemes for conservation laws. In: Lecture Notes in Computational Science and Engineering, vol. 27. Springer, Berlin (2003)

  26. Nave, J.C., Rosales, R., Seibold, B.: A gradient-augmented level set method with an optimally local, coherent advection scheme. J. Comput. Phys. 229(10), 3802–3827 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nguyen van yen, R., Sonnendrücker, E., Schneider, K., Farge, M.: Particle-in-wavelets scheme for the 1D Vlasov–Poisson equations. ESAIM Proc. 32, 134–148 (2011)

    Article  MATH  Google Scholar 

  28. Ottino, J.: The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  29. Plewa, T., Linde, T., Weirs, V.G.: Adaptive mesh refinement: theory and applications. In: Proceedings of the Chicago Workshop on Adaptive Mesh Refinement Methods, Sept. 3–5, 2003, Lecture Notes in Computational Science and Engineering, vol. 41. Springer, Berlin (2005)

  30. Popinet, S.: Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190(2), 572–600 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Popinet, S.: An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228(16), 5838–5866 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rossinelli, D., Hejazialhosseini, B., van Rees, W., Gazzola, M., Bergdorf, M., Koumoutsakos, P.: MRAG-I2D: multi-resolution adapted grids for remeshed vortex methods on multicore architectures. J. Comput. Phys. 288, 1–18 (2015)

    Article  MathSciNet  Google Scholar 

  33. Roussel, O., Schneider, K.: Coherent vortex simulation of weakly compressible turbulent mixing layers using adaptive multiresolution methods. J. Comput. Phys. 229(6), 2267–2286 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Seibold, B., Rosales, R., Nave, J.C.: Jet schemes for advection problems. Discrete Contin. Dyn. Syst. Ser. B 17(4), 1229–1259 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149(2), 201–220 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Staniforth, A., Côté, J.: Semi-Lagrangian integration schemes for atmospheric models: a review. Mon. Weather Rev. 119(9), 2206–2223 (1991)

    Article  Google Scholar 

  38. Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013)

    Book  MATH  Google Scholar 

  39. Wang, S.: Elliptic interface problem solved using the mixed finite element method. Ph.D. thesis, Stony Brook University, (2007)

  40. Warming, R., Beam, R.: Discrete multiresolution analysis using Hermite interpolation: biorthogonal multiwavelets. SIAM J. Sci. Comput. 22(4), 1269–1317 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ziegler, J.L., Deiterding, R., Shepherd, J.E., Pullin, D.I.: An adaptive high-order hybrid scheme for compressive, viscous flows with detailed chemistry. J. Comput. Phys. 230(20), 7598–7630 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

DK acknowledges financial support from the CRM–ISM Fellowship and thanks Alexey Eremin for useful discussions about Runge–Kutta schemes. JCN acknowledges support from the NSERC Discovery and Discovery Accelerator Programs. KS thankfully acknowledges financial support from the ANR project SiCoMHD (ANR-Blanc 2011-045).

Author information

Authors and Affiliations

  1. CRM, 805 Sherbrooke W. Street, Montreal, QC, H3A 0B9, Canada

    Dmitry Kolomenskiy

  2. The Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W. Street, Montreal, QC, H3A 0B9, Canada

    Dmitry Kolomenskiy & Jean-Christophe Nave

  3. M2P2–CNRS, Aix-Marseille Université, 39, rue Frédéric Joliot-Curie, 13453, Marseille Cedex 13, France

    Kai Schneider

Authors
  1. Dmitry Kolomenskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Christophe Nave
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kai Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry Kolomenskiy.

Additional information

This work was supported by the CRM–ISM Fellowship, NSERC Discovery and Discovery Accelerator Programs.

Appendix: Two-Dimensional Hermite Interpolation

Appendix: Two-Dimensional Hermite Interpolation

Suppose that values of a function \(u(x_1,x_2)\) and its derivatives \(u_{x_1}\), \(u_{x_2}\) and \(u_{x_1 x_2}\) are given at four vertices of a square of side \(h\), as shown in Fig. 18. We will use superscripts \(sw\), \(se\), \(nw\) and \(ne\) to refer to these points and the corresponding values.

Fig. 18
figure 18

Interpolation cell. Markers \(\circ \) show the corner points, where the values of the function \(u\) and its derivatives \(u_{x_1}\), \(u_{x_2}\) and \(u_{x_1 x_2}\) are known. Markers \(\times \) show the 3 points that appear in mid-point interpolation formulae (42–45).

Full size image

Let us rescale the coordinates \(x_1\), \(x_2\):

$$\begin{aligned} \tilde{x}_1 = \frac{x_1-x^{\mathrm{sw}}_1}{h}, \quad \quad \tilde{x}_2 = \frac{x_2-x^{\mathrm{sw}}_2}{h}, \end{aligned}$$
(37)

and define basis functions

$$\begin{aligned} f(\tilde{x}) = 2\tilde{x}^3-3\tilde{x}^2+1, \quad \quad g(\tilde{x}) = \tilde{x}^3-2\tilde{x}^2+\tilde{x} \end{aligned}$$
(38)

The value of \(u\) at \((x_1,x_2) \in [x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_1+h]\times [x^{\mathrm{sw}}_2,x^{\mathrm{sw}}_2+h]\) can be estimated using the following \({\mathcal {O}}(h^4)\) accurate formula:

$$\begin{aligned} \tilde{u}(x_1,x_2)= & {} ( u^{\mathrm{sw}} f(\tilde{x}_1) f(\tilde{x}_2) + u^{\mathrm{se}} f(1-\tilde{x}_1) f(\tilde{x}_2) \nonumber \\&+ u^{\mathrm{nw}} f(\tilde{x}_1) f(1-\tilde{x}_2) + u^{\mathrm{ne}} f(1-\tilde{x}_1) f(1-\tilde{x}_2) ) \nonumber \\&+h~ ( u_{x_1}^{\mathrm{sw}} g(\tilde{x}_1) f(\tilde{x}_2) - u_{x_1}^{se} g(1-\tilde{x}_1) f(\tilde{x}_2) \nonumber \\&+ u_{x_1}^{\mathrm{nw}} g(\tilde{x}_1) f(1-\tilde{x}_2) - u_{x_1}^{\mathrm{ne}} g(1-\tilde{x}_1) f(1-\tilde{x}_2) ) \nonumber \\&+h~ ( u_{x_2}^{\mathrm{sw}} f(\tilde{x}_1) g(\tilde{x}_2) + u_{x_2}^{\mathrm{se}} f(1-\tilde{x}_1) g(\tilde{x}_2) \nonumber \\&- u_{x_2}^{\mathrm{nw}} f(\tilde{x}_1) g(1-\tilde{x}_2) - u_{x_2}^{\mathrm{ne}} f(1-\tilde{x}_1) g(1-\tilde{x}_2) ) \nonumber \\&+ h^2~ ( u_{x_1 x_2}^{\mathrm{sw}} g(\tilde{x}_1) g(\tilde{x}_2) - u_{x_1 x_2}^{\mathrm{se}} g(1-\tilde{x}_1) g(\tilde{x}_2) \nonumber \\&- u_{x_1 x_2}^{\mathrm{nw}} g(\tilde{x}_1) g(1-\tilde{x}_2) + u_{x_1 x_2}^{\mathrm{ne}} g(1-\tilde{x}_1) g(1-\tilde{x}_2) ). \end{aligned}$$
(39)

It is straightforward to obtain interpolation formulae for the first and second partial derivatives of \(u\) by derivating (39).

The values \(\tilde{u}(x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2)\), \(\tilde{u}(x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2})\) and \(\tilde{u}(x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2})\), as well as unscaled derivatives required for the error estimate, are also obtained from (39). For multiresolution decomposition (16) and reconstruction (17), we define scaled quantities:

$$\begin{aligned} \begin{array}{llll} u_0^{00} = u^{sw}, &{} u_0^{20} = u^{se}, &{} u_0^{02} = u^{nw}, &{} u_0^{22} = u^{ne}, \\ u_1^{00} = \frac{h}{2} u_{x_1}^{sw}, &{} u_1^{20} = \frac{h}{2} u_{x_1}^{se}, &{} u_1^{02} = \frac{h}{2} u_{x_1}^{nw}, &{} u_1^{22} = \frac{h}{2} u_{x_1}^{ne}, \\ u_2^{00} = \frac{h}{2} u_{x_2}^{sw}, &{} u_2^{20} = \frac{h}{2} u_{x_2}^{se}, &{} u_2^{02} = \frac{h}{2} u_{x_2}^{nw}, &{} u_2^{22} = \frac{h}{2} u_{x_2}^{ne}, \\ u_3^{00} = \frac{h^2}{4} u_{x_1 x_2}^{sw}, &{} u_3^{20} = \frac{h^2}{4} u_{x_1 x_2}^{se}, &{} u_3^{02} = \frac{h^2}{4} u_{x_1 x_2}^{nw}, &{} u_3^{22} = \frac{h^2}{4} u_{x_1 x_2}^{ne}, \end{array} \end{aligned}$$
(40)

as well as

$$\begin{aligned} \begin{array}{ll} \tilde{u}_0^{10} = \tilde{u}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_0^{01} = \tilde{u}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_0^{11} = \tilde{u}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ \tilde{u}_1^{10} = \frac{h}{2} \tilde{u}_{x_1}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_1^{01} = \frac{h}{2} \tilde{u}_{x_1}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_1^{11} = \frac{h}{2} \tilde{u}_{x_1}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ \tilde{u}_2^{10} = \frac{h}{2} \tilde{u}_{x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_2^{01} = \frac{h}{2} \tilde{u}_{x_2}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_2^{11} = \frac{h}{2} \tilde{u}_{x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ \tilde{u}_3^{10} = \frac{h^2}{4} \tilde{u}_{x_1 x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_3^{01} = \frac{h^2}{4} \tilde{u}_{x_1 x_2}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_3^{11} = \frac{h^2}{4} \tilde{u}_{x_1 x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) . \end{array} \end{aligned}$$
(41)

Thus we obtain the following formulae:

$$\begin{aligned} \tilde{u}_0^{10}= & {} \frac{1}{2} \left( u^{00}+u^{10}\right) + \frac{1}{4} \left( u_{x1}^{00}-u_{x1}^{10}\right) , \nonumber \\ \tilde{u}_0^{01}= & {} \frac{1}{2} \left( u^{00}+u^{01}\right) + \frac{1}{4} \left( u_{x2}^{00}-u_{x2}^{01}\right) , \nonumber \\ \tilde{u}_0^{11}= & {} \frac{1}{4} \left( u^{00}+u^{10}+u^{01}+u^{11}\right) \nonumber \\&+ \frac{1}{8} \left( \left( u_{x1}^{00}-u_{x1}^{10}+u_{x1}^{01}-u_{x1}^{11}\right) + \left( u_{x2}^{00}+u_{x2}^{10}-u_{x2}^{01}-u_{x2}^{11}\right) \right) \nonumber \\&+ \frac{1}{16} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{10}-u_{x_1 x_2}^{01}+u_{x_1 x_2}^{11}\right) ,\end{aligned}$$
(42)
$$\begin{aligned} \tilde{u}_1^{10}= & {} - \frac{3}{4} \left( u^{00}-u^{10}\right) - \frac{1}{4} \left( u_{x_1}^{00}+u_{x_1}^{10}\right) , \nonumber \\ \tilde{u}_1^{01}= & {} \frac{1}{2} \left( u_{x_1}^{00}+u_{x_1}^{01}\right) + \frac{1}{4} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{01}\right) , \nonumber \\ \tilde{u}_1^{11}= & {} \frac{3}{8} \left( -u^{00}+u^{10}-u^{01}+u^{11}\right) \nonumber \\&- \frac{1}{8} \left( u_{x_1}^{00}+u_{x_1}^{10}+u_{x_1}^{01}+u_{x_1}^{11}\right) - \frac{3}{16} \left( u_{x_2}^{00}-u_{x_2}^{10}-u_{x_2}^{01}+u_{x_2}^{11}\right) \nonumber \\&- \frac{1}{16} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{10}-u_{x_1 x_2}^{01}-u_{x_1 x_2}^{11}\right) \end{aligned}$$
(43)
$$\begin{aligned} \tilde{u}_2^{10}= & {} \frac{1}{2} \left( u_{x_2}^{00}+u_{x_2}^{10}\right) + \frac{1}{4} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{10}\right) , \nonumber \\ \tilde{u}_2^{01}= & {} - \frac{3}{4} \left( u^{00}-u^{01}\right) - \frac{1}{4} \left( u_{x_2}^{00}+u_{x_2}^{01}\right) , \nonumber \\ \tilde{u}_2^{11}= & {} \frac{3}{8} \left( -u^{00}-u^{10}+u^{01}+u^{11}\right) \nonumber \\&- \frac{3}{16} \left( u_{x_1}^{00}-u_{x_1}^{10}-u_{x_1}^{01}+u_{x_1}^{11}\right) - \frac{1}{8} \left( u_{x_2}^{00}+u_{x_2}^{10}+u_{x_2}^{01}+u_{x_2}^{11}\right) \nonumber \\&- \frac{1}{16} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{10}+u_{x_1 x_2}^{01}-u_{x_1 x_2}^{11}\right) \end{aligned}$$
(44)
$$\begin{aligned} \tilde{u}_3^{10}= & {} - \frac{3}{4} \left( u_{x_2}^{00}-u_{x_2}^{10}\right) - \frac{1}{4} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{10}\right) , \nonumber \\ \tilde{u}_3^{01}= & {} - \frac{3}{4} \left( u_{x_1}^{00}-u_{x_1}^{01}\right) - \frac{1}{4} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{01}\right) , \nonumber \\ \tilde{u}_3^{11}= & {} \frac{9}{16} \left( u^{00}-u^{10}-u^{01}+u^{11}\right) \nonumber \\&+ \frac{3}{16} \left( \left( u_{x_1}^{00}+u_{x_1}^{10}-u_{x_1}^{01}-u_{x_1}^{11}\right) + \left( u_{x_2}^{00}-u_{x_2}^{10}+u_{x_2}^{01}-u_{x_2}^{11}\right) \right) \nonumber \\&+ \frac{1}{16} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{10}+u_{x_1 x_2}^{01}+u_{x_1 x_2}^{11}\right) \end{aligned}$$
(45)

In the notations of (16–17), we obtain

$$\begin{aligned} (\tilde{u}_\iota )_{2j_1+1,2j_2}^l = \tilde{u}_\iota ^{10}, \quad (\tilde{u}_\iota )_{2j_1,2j_2+1}^l = \tilde{u}_\iota ^{01}, \quad (\tilde{u}_\iota )_{2j_1+1,2j_2+1}^l = \tilde{u}_\iota ^{11}, \end{aligned}$$
(46)

where \(\iota = 0,\ldots ,3\) and indices \(j_1\), \(j_2\) and \(l\) are determined by \(\varvec{x}^{sw}\) and \(h\), as described in Sects. 2.2 and 2.3. Note that the computational cost of this procedure is much less than interpolation at an arbitrary point because the coefficients that include the basis functions (38) are pre-computed analytically.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kolomenskiy, D., Nave, JC. & Schneider, K. Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation. J Sci Comput 66, 116–140 (2016). https://doi.org/10.1007/s10915-015-0014-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0014-7

Keywords

Mathematics Subject Classification

Search

Navigation