Applications of the Newton-Raphson method in a SDFEM for inviscid Burgers equation. (English) Zbl 1474.76038
Summary: In this paper we investigate in detail the applications of the classical Newton-Raphson method in connection with a space-time finite element discretization scheme for the inviscid Burgers equation in one dimensional space. The underlying discretization method is the so-called streamline diffusion method, which combines good stability properties with high accuracy. The coupled nonlinear algebraic equations thus obtained in each space-time slab are solved by the generalized Newton-Raphson method. Exploiting the band-structured properties of the Jacobian matrix, two different algorithms based on the Newton-Raphson linearization are proposed. In a series of examples, we show that in each time-step a quadratic convergence order is attained when the Newton-Raphson procedure applied to the corresponding system of nonlinear equations.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 65H10 | Numerical computation of solutions to systems of equations |
References:
| [1] | M. Asadzadeh and P. Kowalczyk,Convergence of streamline diffusion methods for the Vlasov-PoissonFokker-Planck system, Numer. Methods Partial Differential Equations,21(3) (2005), 472-495. · Zbl 1072.82023 |
| [2] | L. Beilina and C. Johnson,A posteriori error estimation in computational inverse scattering, Math. Models Methods Appl. Sci,15(1) (2005), 23-36. · Zbl 1160.76363 |
| [3] | N. Bicanic and K. Johnson,Who was Raphson?, Int. J. Numer. Methods. Eng.,14(1) (1979), 148-152. · Zbl 0398.01005 |
| [4] | J. M. Burgers,A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., (1948), 171-199. |
| [5] | P. G. Ciarlet,The Finite Element Method for Elliptic Problems, Amesterdam, North Holland, 1987. |
| [6] | E. Godlewski and P. A. Raviart,Numerical Approximation of Hyperbolic Systems of Conservation laws, Vol. 118. Springer Science and Business Media, 1996. · Zbl 0860.65075 |
| [7] | P. Hansbo,The characteristic streamline diffusion method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg,96(2) (1992), 239-253. · Zbl 0753.76095 |
| [8] | H. Hartmann,A remark on the application of the Newton-Raphson method in non-linear finite element analysis, Comput. Mech,36(2) (2005), 100-116. · Zbl 1102.74040 |
| [9] | E. Hopf,The partial differential equationut+uux=µuxx, Comm. Pure Appl. Math,3(3) (1950), 201-230. · Zbl 0039.10403 |
| [10] | T. J. R. Hughes and A. Brooks,In: T.J.R. Hughes (eds.), Finite Element methods for Convection Dominated flows, AMD-vol-34 (1979), 19-35. · Zbl 0423.76067 |
| [11] | T. J. R. Hughes and A. Brooks,Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg.,32(1-3) (1982), 199-259. · Zbl 0497.76041 |
| [12] | M. Izadi, Streamline diffusion methods for treating the coupling equations of two hyperbolic conservation laws, Math. Comput. Model,45(1-2) (2007), 201-214. · Zbl 1133.65076 |
| [13] | M. Izadi,A posteriori error estimates for the coupling equations of scalar conservation laws, BIT. Numer. Math.,49(4) (2009), 697-720. · Zbl 1190.65138 |
| [14] | C. Johnson,Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg,107(1-2) (1993), 117-129. · Zbl 0787.65070 |
| [15] | C. Johnson, U. N¨avert, and J. Pitk¨aranta,Finite element method for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg,45(1984), 285-312. · Zbl 0526.76087 |
| [16] | C. Johnson and J. Saranen,Streamline diffusion methods for the incompressible Euler and NavierStokes equations, Math. Comp.,47(175) (1986), 1-18. · Zbl 0609.76020 |
| [17] | C. Johnson and A. Szepessy,On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp.,49(180) (1987), 427-444. · Zbl 0634.65075 |
| [18] | C. Johnson and A. Szepessy, In: Tezduyar, T.E., Hughes, T.J.R., (eds.),Numerical Methods for Compressible Flow-Finite Difference, Element and Volume Techniques, AMD-vol-78, 1986. |
| [19] | C. Johnson, A. Szepessy, and P. Hansbo,On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp,54(189) (1990), 107-129. · Zbl 0685.65086 |
| [20] | S. N. Kruzhkov,First order quasilinear equations in several independent variables, USSR Math. Sbornik.10(2) (1970), 217-243. · Zbl 0215.16203 |
| [21] | R. Sandboge,Adaptive Finite Element Methods for Reactive Flow Problems, Ph.D. Thesis, Department of Mathematics, Chalmers University of Technology, G¨oteborg, 1996. · Zbl 0944.76038 |
| [22] | A. |
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.
