Conservative high precision pseudo arc-length method for strong discontinuity of detonation wave

doi:10.1007/s10483-022-2817-9

Abstract

Abstract: A hyperbolic conservation equation can easily generate strong discontinuous solutions such as shock waves and contact discontinuity. By introducing the arc-length parameter, the pseudo arc-length method (PALM) smoothens the discontinuous solution in the arc-length space. This in turn weakens the singularity of the equation. To avoid constructing a high-order scheme directly in the deformed physical space, the entire calculation process is conducted in a uniform orthogonal arc-length space. Furthermore, to ensure the stability of the equation, the time step is reduced by limiting the moving speed of the mesh. Given that the calculation does not involve the interpolation process of physical quantities after the mesh moves, it maintains a high computational efficiency. The numerical examples show that the algorithm can effectively reduce numerical oscillations while maintaining excellent characteristics such as high precision and high resolution.

Key words: pseudo arc-length method (PALM), conservative, strong discontinuity, high-order, weighted essentially non-oscillatory (WENO)

2010 MSC Number:

35L55
35L65

Tianbao MA, Chentao WANG, Xiangzhao XU. Conservative high precision pseudo arc-length method for strong discontinuity of detonation wave. Applied Mathematics and Mechanics (English Edition), 2022, 43(3): 417-436.

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References

[1] NING, J. G., LI, J. Q., and SONG, W. D. Investigation of plasma damage properties generated by hypervelocity impact. Chinese Journal of Theoretical and Applied Mechanics, 46(6), 853–861(2014)
[2] NING, J. G. and CHEN, L. W. Fuzzy interface treatment in Eulerian method. Science in China Series E: Technological Sciences, 47, 550–568(2004)
[3] NING, J. G., REN, H. L., GUO, T. T., and LI, P. Dynamic response of alumina ceramics impacted by long tungsten projectile. International Journal of Impact Engineering, 62, 60–74(2013)
[4] RIKS, E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures, 15(7), 529–551(1979)
[5] BOOR, C. On local spline approximation by moments. Journal of Mathematical Mechanics, 17(1), 729–735(1968)
[6] CHAN, T. F. Newton-like pseudo-arclength methods for computing simple turning points. SIAM Journal on Scientific and Statistical Computing, 5(1), 135–148(1984)
[7] XIN, X. K. and TANG, D. H. An arc-length parameter technique for steady diffusion-convection equation. Chinese Journal of Applied Mechanics, 5(1), 45–54(1988)
[8] WU, J. K., XU, W. H., and DING, H. L. Arc-length method for differential equations. Applied Mathematics and Mechanics (English Edition), 20(8), 875–880(1999) https://doi.org/10.1007/BF02452494
[9] WINSLOW, A. M. Numerical solution of the quasi-linear Poisson equation. Journal of Computational Physics, 1, 149–172(1967)
[10] BRACKBILL, J. U. and SALTZMAN, J. S. Adaptive zoning for singular problems in two dimensions. Journal of Computational Physics, 46(3), 342–368(1982)
[11] CHEN, K. Error equidistribution and mesh adaptation. SIAM Journal on Scientific Computing, 15(4), 798–818(1994)
[12] TANG, H. Z. and TANG, T. Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 41(2), 487–515(2003)
[13] XU, X., DAI, Z. H., and GAO, Z. M. A 3D cell-centered Lagrangian scheme for the ideal magnetohydrodynamics equations on unstructured meshes. Computer Methods in Applied Mechanics and Engineering, 342, 490–508(2018)
[14] MARTI, J. and RYZHAKOV, P. B. An explicit-implicit finite element model for the numerical solution of incompressible Navier-Stokes equations on moving grids. Computer Methods in Applied Mechanics and Engineering, 350, 750–765(2019)
[15] ZHANG, M., HUANG, W. Z., and QIU, J. X. A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography. Journal of Scientific Computing, 87, 88(2021)
[16] NING, J. G., YUAN, X. P., MA, T. B., and LI, J. Pseudo arc-length numerical algorithm for computational dynamics. Chinese Journal of Theoretical and Applied Mechanics, 49(3), 703–715(2017)
[17] NING, J. G., YUAN, X. P., MA, T. B., and WANG, C. Positivity-preserving moving mesh scheme for two-step reaction model in two dimensions. Computers and Fluids, 123, 72–86(2015)
[18] WANG, X., MA, T. B., REN, H. L., and NING, J. G. A local pseudo arc-length method for hyperbolic conservation laws. Acta Mechanica Sinica, 30(6), 956–965(2014)
[19] YUAN, X. P., NING, J. G., MA, T. B., and WANG, C. Stability of Newton TVD Runge-Kutta scheme for one-dimensional Euler equations with adaptive mesh. Applied Mathematics and Computation, 282, 1–16(2016)
[20] ZHANG, H., WANG, G., and ZHANG, F. A multi-resolution weighted compact nonlinear scheme for hyperbolic conservation laws. International Journal of Computational Fluid Dynamics, 34(3), 187–203(2020)
[21] ZHENG, S. C., DENG, X. G., and WANG, D. F. New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications. Applied Mathematics and Mechanics (English Edition), 42(3), 405–424(2021) https://doi.org/10.1007/s10483-021- 2712-8
[22] JIANG, G. S. and SHU, C. W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1), 202–228(1996)
[23] ZHANG, Y., DENG, W. H., and ZHU, J. A new sixth-order finite difference WENO scheme for fractional differential equations. Journal of Scientific Computing, 87, 73(2021)
[24] LEI, N., CHENG, J., and SHU, C. W. A high order positivity-preserving conservative WENO remapping method on 2D quadrilateral meshes. Computer Methods in Applied Mechanics and Engineering, 3763, 113497(2021)
[25] ZHENG, F. and QIU, J. X. Directly solving the Hamilton-Jacobi equations by Hermite WENO schemes. Journal of Computational Physics, 307, 423–445(2016)
[26] LI, M. J., YANG, Y. Y., and SHU, S. Third-order modified coefficient scheme based on essentially non-oscillatory scheme. Applied Mathematics and Mechanics (English Edition), 29(11), 1477–1486(2008) https://doi.org/10.1007/s10483-008-1108-x
[27] WANG, Y. and YUAN, L. A new 6th-order WENO scheme with modified stencils. Computers and Fluids, 208, 104625(2020)
[28] LIU, S. P., SHEN, Y. Q., PENG, J., and ZHANG, J. Two-step weighting method for constructing fourth-order hybrid central WENO scheme. Computers and Fluids, 207, 104590(2020)
[29] GANDE, N. R. and BHISE, A. A. Third-order WENO schemes with kinetic flux vector splitting. Applied Mathematics and Computation, 378, 125203(2020)
[30] WIBISONO, I., YANUAR, and KOSASIH, E. A. Fifth-order Hermite targeted essentially nonoscillatory schemes for hyperbolic conservation laws. Journal of Scientific Computing, 87, 69(2021)
[31] YANG, X. B., HUANG, W. Z., and QIU, J. X. A moving mesh method for one-dimensional conservation laws. SIAM Journal on Scientific Computing, 34(4), 2317–2343(2012)
[32] LI, Y., CHENG, J., XIA, Y., and SHU, C. W. On moving mesh WENO schemes with characteristic boundary conditions for Hamilton-Jacobi equations. Computers and Fluids, 205, 104582(2020)
[33] CHRISTLIEB, A., GUO, W., JIANG, Y., and YANG, H. A moving mesh WENO method based on exponential polynomials for one-dimensional conservation laws. Journal of Computational Physics, 380, 334–354(2019)
[34] DENG, X. G., MAO, M. L., TU, G. H., LIU, H. Y., and ZHANG, H. X. Geometric conservation law and applications to high-order finite difference schemes with stationary grids. Journal of Computational Physics, 230(4), 1100–1115(2011)
[35] ZHU, Z. B., YANG, W. B., and YU, M. A WENO scheme with geometric conservation law and its application. Chinese Journal of Computational Mechanics, 34(6), 779–784(2017)
[36] XU, D., WANG, D. F., CHEN, Y. M., and DENG, X. G. High-order finite volume schemes in three-dimensional curvilinear coordinate system. Journal of National University of Defense Technology, 38(2), 56–60(2016)
[37] LIU, J., LIU, Y., and CHEN, Z. D. Unstructured deforming mesh and discrete geometric conservation law. Acta Aeronautica et Astronautica Sinica, 37(8), 2395–2407(2016)
[38] LIU, J., HAN, F., and XIA, B. Mechanism and algorithm for geometric conservation law in finite difference method. Acta Aerodynamica Sinica, 36(6), 917–926(2018)
[39] BENOIT, M. and NADARAJAH, S. On the geometric conservation law for the non-linear frequency domain and time-spectral methods. Computer Methods in Applied Mechanics and Engineering, 355, 690–728(2019)
[40] YAN, C. Computational Fluid Dynamic’s Methods and Applications, Beihang University Press, Beijing, 15–25(2006)
[41] TAO, Z. J., LI, F. Y., and QIU, J. X. High-order central Hermite WENO schemes: dimensionby-dimension moment-based reconstructions. Journal of Computational Physics, 318, 222–251(2016)
[42] TAO, Z. J. and QIU, J. X. Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations. Advances in Computational Mathematics, 43(5), 1023–1058(2017)
[43] CHEN, R. S. Computation of compressible flows with high density ratio and pressure ratio. Applied Mathematics and Mechanics (English Edition), 29(5), 673–682(2008) https://doi.org/10.1007/s10483-008-0511-y
[44] TANG, L. Y., SONG, S. H., and ZHANG, H. High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws. Applied Mathematics and Mechanics (English Edition), 41(1), 173–192(2020) https://doi.org/10.1007/s10483-020-2554-8
[45] GONG, S. and WU, C. J. A study of a supersonic capsule/rigid disk-gap-band parachute system using large-eddy simulation. Applied Mathematics and Mechanics (English Edition), 42(4), 485–500(2021) https://doi.org/10.1007/s10483-021-2716-5
[46] KUZIMIN, D. Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 361, 112804(2020)