×

Conservative high precision pseudo arc-length method for strong discontinuity of detonation wave. (English) Zbl 1496.65199

Summary: 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.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L55 Higher-order hyperbolic systems
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35Q31 Euler equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ning, J. G.; Li, J. Q.; 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.; 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.; 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) · Zbl 0408.73040
[5] Boor, C., On local spline approximation by moments, Journal of Mathematical Mechanics, 17, 1, 729-735 (1968) · Zbl 0162.08402
[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) · Zbl 0536.65029
[7] Xin, X. K.; 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.; Ding, H. L., Arc-length method for differential equations, Applied Mathematics and Mechanics (English Edition), 20, 8, 875-880 (1999)
[9] Winslow, A. M., Numerical solution of the quasi-linear Poisson equation, Journal of Computational Physics, 1, 149-172 (1967) · Zbl 0254.65069
[10] Brackbill, J. U.; Saltzman, J. S., Adaptive zoning for singular problems in two dimensions, Journal of Computational Physics, 46, 3, 342-368 (1982) · Zbl 0489.76007
[11] Chen, K., Error equidistribution and mesh adaptation, SIAM Journal on Scientific Computing, 15, 4, 798-818 (1994) · Zbl 0806.65090
[12] Tang, H. Z.; Tang, T., Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 41, 2, 487-515 (2003) · Zbl 1052.65079
[13] Xu, X.; Dai, Z. H.; 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) · Zbl 1440.76101
[14] Marti, J.; 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) · Zbl 1441.76065
[15] Zhang, M.; Huang, W. Z.; 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) · Zbl 1476.65226
[16] Ning, J. G.; Yuan, X. P.; Ma, T. B.; 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.; Wang, C., Positivity-preserving moving mesh scheme for two-step reaction model in two dimensions, Computers and Fluids, 123, 72-86 (2015) · Zbl 1390.80021
[18] Wang, X.; Ma, T. B.; Ren, H. L.; Ning, J. G., A local pseudo arc-length method for hyperbolic conservation laws, Acta Mechanica Sinica, 30, 6, 956-965 (2014) · Zbl 1342.65177
[19] Yuan, X. P.; Ning, J. G.; Ma, T. B.; 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) · Zbl 1410.65266
[20] Zhang, H.; Wang, G.; Zhang, F., A multi-resolution weighted compact nonlinear scheme for hyperbolic conservation laws, International Journal of Computational Fluid Dynamics, 34, 3, 187-203 (2020) · Zbl 1483.65143
[21] Zheng, S. C.; Deng, X. G.; 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) · Zbl 1479.76072
[22] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126, 1, 202-228 (1996) · Zbl 0877.65065
[23] Zhang, Y.; Deng, W. H.; Zhu, J., A new sixth-order finite difference WENO scheme for fractional differential equations, Journal of Scientific Computing, 87, 73 (2021) · Zbl 1468.65120
[24] Lei, N.; Cheng, J.; 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) · Zbl 1506.65126
[25] Zheng, F.; Qiu, J. X., Directly solving the Hamilton-Jacobi equations by Hermite WENO schemes, Journal of Computational Physics, 307, 423-445 (2016) · Zbl 1354.65179
[26] Li, M. J.; Yang, Y. Y.; Shu, S., Third-order modified coefficient scheme based on essentially non-oscillatory scheme, Applied Mathematics and Mechanics (English Edition), 29, 11, 1477-1486 (2008) · Zbl 1176.65092
[27] Wang, Y.; Yuan, L., A new 6th-order WENO scheme with modified stencils, Computers and Fluids, 208, 104625 (2020) · Zbl 1502.65077
[28] Liu, S. P.; Shen, Y. Q.; Peng, J.; Zhang, J., Two-step weighting method for constructing fourth-order hybrid central WENO scheme, Computers and Fluids, 207, 104590 (2020) · Zbl 1521.76560
[29] Gande, N. R.; Bhise, A. A., Third-order WENO schemes with kinetic flux vector splitting, Applied Mathematics and Computation, 378, 125203 (2020) · Zbl 1508.76077
[30] Wibisono, I.; Yanuar; Kosasih, E. A., Fifth-order Hermite targeted essentially non-oscillatory schemes for hyperbolic conservation laws, Journal of Scientific Computing, 87, 69 (2021) · Zbl 1465.65087
[31] Yang, X. B.; Huang, W. Z.; Qiu, J. X., A moving mesh method for one-dimensional conservation laws, SIAM Journal on Scientific Computing, 34, 4, 2317-2343 (2012) · Zbl 1253.65133
[32] Li, Y.; Cheng, J.; Xia, Y.; Shu, C. W., On moving mesh WENO schemes with characteristic boundary conditions for Hamilton-Jacobi equations, Computers and Fluids, 205, 104582 (2020) · Zbl 1519.65020
[33] Christlieb, A.; Guo, W.; Jiang, Y.; Yang, H., A moving mesh WENO method based on exponential polynomials for one-dimensional conservation laws, Journal of Computational Physics, 380, 334-354 (2019) · Zbl 1451.65106
[34] Deng, X. G.; Mao, M. L.; Tu, G. H.; Liu, H. Y.; 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) · Zbl 1210.65153
[35] Zhu, Z. B.; Yang, W. B.; Yu, M., A WENO scheme with geometric conservation law and its application, Chinese Journal of Computational Mechanics, 34, 6, 779-784 (2017) · Zbl 1399.76031
[36] Xu, D.; Wang, D. F.; Chen, Y. M.; 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.; 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.; Xia, B., Mechanism and algorithm for geometric conservation law in finite difference method, Acta Aerodynamica Sinica, 36, 6, 917-926 (2018)
[39] Benoit, M.; 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) · Zbl 1441.76087
[40] Yan, C., Computational Fluid Dynamic’s Methods and Applications, 15-25 (2006), Beijing: Beihang University Press, Beijing
[41] Tao, Z. J.; Li, F. Y.; Qiu, J. X., High-order central Hermite WENO schemes: dimension-by-dimension moment-based reconstructions, Journal of Computational Physics, 318, 222-251 (2016) · Zbl 1349.65385
[42] Tao, Z. J.; 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) · Zbl 1378.65162
[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) · Zbl 1231.65169
[44] Tang, L. Y.; Song, S. H.; 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) · Zbl 1462.35199
[45] Gong, S.; 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)
[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) · Zbl 1442.65263
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.