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Adaptive gradient-augmented level set method with multiresolution error estimation. (English) Zbl 1338.35290
Summary: 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.

MSC:
35L65 Hyperbolic conservation laws
35Q35 PDEs in connection with fluid mechanics
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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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) · Zbl 1433.76144
[2] Appelö, D; Hagstrom, T, On advection by Hermite methods, Pac. J. Appl. Math., 4, 125-139, (2011) · Zbl 1295.65099
[3] Becker, R; Rannacher, R, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10, 1-102, (2001) · Zbl 1105.65349
[4] Berger, MJ; Colella, P, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64-84, (1989) · Zbl 0665.76070
[5] Berger, M; LeVeque, R, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J. Numer. Anal., 35, 2298-2316, (1998) · Zbl 0921.65070
[6] Berger, MJ; Oliger, J, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 484-512, (1984) · Zbl 0536.65071
[7] Brun, E; Guittet, A; Gibou, F, A local level-set method using a hash table data structure, J. Comput. Phys., 231, 2528-2536, (2012) · Zbl 1242.65156
[8] Chiavassa, G; Donat, R, Point value multiscale algorithms for 2D compressible flows, SIAM J. Sci. Comput., 23, 805-823, (2001) · Zbl 1043.76046
[9] Chidyagwal, P; Nave, JC; Rosales, R; Seibold, B, A comparative study of the efficiency of jet schemes, Int. J. Numer. Anal. Model. Ser. B, 3, 297-306, (2012) · Zbl 1260.65089
[10] Cohen, A; Ciarlet, P (ed.); Lions, J (ed.), Wavelet methods in numerical analysis, No. 7, 417-711, (2000), Amsterdam · Zbl 0976.65124
[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) · Zbl 1365.76148
[12] Deiterding, R, Block-structured adaptive mesh refinement: theory, implementation and application, ESAIM Proc., 34, 97-150, (2011) · Zbl 1302.65220
[13] DeVore, R, Nonlinear approximation, Acta Numer., 7, 51-150, (1998) · Zbl 0931.65007
[14] Domingues, M; Gomes, S; Roussel, O; Schneider, K, An adaptive multiresolution scheme with local time stepping for evolutionary pdes, J. Comput. Phys., 227, 3758-3780, (2008) · Zbl 1139.65060
[15] Domingues, M; Gomes, S; Roussel, O; Schneider, K, Adaptive multiresolution methods, ESAIM Proc., 34, 1-96, (2011) · Zbl 1302.65185
[16] Dormand, JR; Prince, PJ, New Runge-Kutta algorithms for numerical simulation in dynamical astronomy, Celest. Mech., 18, 223-232, (1978) · Zbl 0386.70006
[17] Harten, A, Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Commun. Pure Appl. Math., 48, 1305-1342, (1995) · Zbl 0860.65078
[18] Harten, A, Multiresolution representation of data: a general framework, SIAM J. Numer. Anal., 33, 1205-1256, (1996) · Zbl 0861.65130
[19] Kaibara, M; Gomes, S; Toro, E (ed.), A fully adaptive multiresolution scheme for shock computations, 503-597, (2001), Dordrecht/New York · Zbl 1064.76589
[20] Lindsay, K; Krasny, R, A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow, J. Comput. Phys., 172, 879-907, (2001) · Zbl 1002.76093
[21] Liu, XD; Osher, S; Chan, T, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 200-212, (1994) · Zbl 0811.65076
[22] Losasso, F; Gibou, F; Fedkiw, R, Simulating water and smoke with an octree data structure, ACM Trans. Graph. TOG, 23, 457-462, (2004)
[23] Martin, DF; Colella, P, A cell-centered adaptive projection method for the incompressible Euler equations, J. Comput. Phys., 163, 271-312, (2000) · Zbl 0991.76052
[24] Min, C; Gibou, F, A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys., 225, 300-321, (2007) · Zbl 1122.65077
[25] Müller, S.: Adaptive multiscale schemes for conservation laws. In: Lecture Notes in Computational Science and Engineering, vol. 27. Springer, Berlin (2003) · Zbl 1302.65185
[26] Nave, JC; Rosales, R; Seibold, B, A gradient-augmented level set method with an optimally local, coherent advection scheme, J. Comput. Phys., 229, 3802-3827, (2010) · Zbl 1189.65214
[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) · Zbl 1301.76063
[28] Ottino, J.: The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press, Cambridge (1989) · Zbl 0721.76015
[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) · Zbl 1053.65002
[30] Popinet, S, Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. Comput. Phys., 190, 572-600, (2003) · Zbl 1076.76002
[31] Popinet, S, An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys., 228, 5838-5866, (2009) · Zbl 1280.76020
[32] Rossinelli, D; Hejazialhosseini, B; 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) · Zbl 1351.76026
[33] Roussel, O; Schneider, K, Coherent vortex simulation of weakly compressible turbulent mixing layers using adaptive multiresolution methods, J. Comput. Phys., 229, 2267-2286, (2010) · Zbl 1303.76137
[34] Seibold, B; Rosales, R; Nave, JC, Jet schemes for advection problems, Discrete Contin. Dyn. Syst. Ser. B, 17, 1229-1259, (2012) · Zbl 1252.65158
[35] Shu, CW; Osher, S, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072
[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, 201-220, (1999) · Zbl 0934.76073
[37] Staniforth, A; Côté, J, Semi-Lagrangian integration schemes for atmospheric models: a review, Mon. Weather Rev., 119, 2206-2223, (1991)
[38] Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013) · Zbl 1279.65127
[39] Wang, S.: Elliptic interface problem solved using the mixed finite element method. Ph.D. thesis, Stony Brook University, (2007) · Zbl 1295.65099
[40] Warming, R; Beam, R, Discrete multiresolution analysis using Hermite interpolation: biorthogonal multiwavelets, SIAM J. Sci. Comput., 22, 1269-1317, (2000) · Zbl 0977.42023
[41] Ziegler, JL; Deiterding, R; Shepherd, JE; Pullin, DI, An adaptive high-order hybrid scheme for compressive, viscous flows with detailed chemistry, J. Comput. Phys., 230, 7598-7630, (2011) · Zbl 1433.76144
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