Convergence analysis of some first order and second order time accurate gradient schemes for semilinear second order hyperbolic equations

This work is devoted to the convergence analysis of finite volume schemes for a model of semilinear second order hyperbolic equations. The model includes for instance the so‐called Sine‐Gordon equation which appears for instance in Solid Physics (cf. Fang and Li, Adv Math (China) 42 (2013), 441–457; Liu et al., Numer Methods Partial Differ Equ 31 (2015), 670–690). We are motivated by two works. The first one is Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043) where a recent class of nonconforming finite volume meshes is introduced. The second one is Eymard et al. (Numer Math 82 (1999), 91–116) where a convergence of a finite volume scheme for semilinear elliptic equations is provided. The mesh considered in Eymard et al. (Numer Math 82 (1999), 91–116) is admissible in the sense of Eymard et al. (Elsevier, Amsterdam, 2000, 723–1020) and a convergence of a family of approximate solutions toward an exact solution when the mesh size tends to zero is proved. This article is also a continuation of our previous two works (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321; Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39) which dealt with the convergence analysis of implicit finite volume schemes for the wave equation. We use as discretization in space the generic spatial mesh introduced in Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043), whereas the discretization in time is performed using a uniform mesh. Two finite volume schemes are derived using the discrete gradient of Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043). The unknowns of these two schemes are the values at the center of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The first scheme is inspired from the previous work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39), whereas the second one (in which the discretization in time is performed using a Newmark method) is inspired from the work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321). Under the assumption that the mesh size of the time discretization is small, we prove the existence and uniqueness of the discrete solutions. If we assume in addition to this that the exact solution is smooth, we derive and prove three error estimates for each scheme. The first error estimate is concerning an estimate for the error between a discrete gradient of the approximate solution and the gradient of the exact solution whereas the second and the third ones are concerning the estimate for the error between the exact solution and the discrete solution in the discrete seminorm of and in the norm of . The convergence rate is proved to be for the first scheme and for the second scheme, where (resp. k) is the mesh size of the spatial (resp. time) discretization. The existence, uniqueness, and convergence results stated above do not require any relation between k and . The analysis presented in this work is also applicable in the gradient schemes framework. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 5–33, 2017     

ADI quadratic finite volume element methods for second order hyperbolic problems

We use the biquadratic elements to develop an alternating direction implicit (ADI) finite volume element method for second order hyperbolic problems in two spatial dimensions. The optimal H ¹-norm error estimate of second order accuracy is proved. Numerical experiments that corroborate the theoretical analysis are also presented.     

High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients

In this paper, we develop implicit difference schemes of O(k⁴ + k²h² + h⁴), where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions. The proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable ADI method of O(k⁴ + k²h² + h⁴). Some physical nonlinear problems are provided to demonstrate the accuracy of the implementation.     

Circulant preconditioners for second order hyperbolic equations

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO() orO(m), where is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.     

Efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles

Summary. Piecewise Hermite bicubic orthogonal spline collocation Laplace-modified and alternating-direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed. The schemes are shown to be unconditionally stable and of optimal order accuracy in the and discrete maximum norms for space and time, respectively. Implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms. Received November 7, 1994 / Revised version received April 29, 1996     

Optimized higher order time discretization of second order hyperbolic problems: Construction and numerical study

We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail. The optimal schemes are validated through various numerical results.     

A selfadjoint second order hyperbolic system

We consider a second order hyperbolic system of the type

(1)     

Oblique derivative problems for second order nonlinear equations of mixed type with degenerate hyperbolic curve

The present paper deals with oblique derivative problems for second order nonlinear equations of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point principle. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.     

Conditions of solvability of quasilinear periodic boundary-value problems for hyperbolic equations of the second order

On the basis of properties of the Vejvoda-Shtedry operator, we obtain solvability conditions for the 2π-periodic problem

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Explicit finite element schemes for first order symmetric hyperbolic systems

Summary Finite element methods of up to fourth order accuracy admitting explicit discrete equations are constructed for linear symmetric first order hyperbolic equations having sufficiently smooth solutions. Lumping of the mass matrix at the forward time level is achieved by the addition of a differential operator, which for smooth spline spaces is dissipative and strongly enhances the stability properties of the resulting scheme.     

Gaussian spectral rules for second order finite-difference schemes

Earlier the authors suggested an algorithm of grid optimization for a second order finite-difference approximation of a two-point problem. The algorithm yields exponential superconvergence of the Neumann-to-Dirichlet map (or the boundary impedance). Here we extend that approach to PDEs with piecewise-constant coefficients and rectangular homogeneous subdomains. Examples of the numerical solution of the 2-dimensional oscillatory Helmholtz equation exhibit exponential convergence at prescribed points, where the cost per grid node is close to that of the standard five-point finite-difference scheme. Our scheme demonstrates high accuracy with slightly more than two grid points per wavelength and reduces the grid size by more than three orders as compared with the standard scheme.     

Blow-up examples for second order elliptic PDEs of critical Sobolev growth

Let be a smooth compact Riemannian manifold of dimension , and be the Laplace-Beltrami operator. Let also be the critical Sobolev exponent for the embedding of the Sobolev space into Lebesgue's spaces, and be a smooth function on . Elliptic equations of critical Sobolev growth such as

have been the target of investigation for decades. A very nice -theory for the asymptotic behaviour of solutions of such equations has been available since the 1980's. The -theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of . It was used as a key point by Druet to prove compactness results for equations such as . An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of . We present such examples in this article.

     

What are second order hyperbolic systems?

On adopting the usual view that a second order differential system is hyperbolic provided that the equivalent first order one is, a theorem is proved which offers an intrinsic procedure to check hyperbolicity of second order systems. In deducing such result, the link between second order and first order systems is analyzed in detail and a number of relevant results is proved.     

Theory of degenerate second order hyperbolic equations

Yakutsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 2, pp. 68–75, March–April, 1990.     

Symmetric hyperbolic difference schemes and matrix problems

Sufficient conditions for the stability of multidimensional finite difference schemes are difficult to obtain. It is shown that for special families of amplification matrices G (A, B) a sufficient condition for power boundedness can be obtained by replacing the matrices by appropriate scalars, and so the problem is reduced to a scalar one. As one application it is shown that the Lax-Wendroff scheme in two dimensions is stable if |Au|^{$\text{2}\text{3}$} + |Bu|^{$\text{2}\text{3}$} ? 1 for all real unit vectors u. The Lax- Wendroff scheme with stabilizer does not always permit such large time steps. It is conjectured that the analysis for all symmetric hyperbolic schemes can be reduced to the scalar case.