Nonlinear Second-order Hyperbolic Equations Research Articles

On a nonlinear hyperbolic equation with a bistable reaction term

Nonlinear second-order hyperbolic equations are gaining ground as models in many areas of application, as extensions of parabolic reaction–diffusion equations that might otherwise be used. The theory of travelling-wave solutions of such reaction–diffusion equations is well established. The present paper is concerned with its counterpart for the wider class of equations in the particular case that the reaction term is bistable. Conditions that are necessary and sufficient for the existence and uniqueness of these solutions are determined. A combination of traditional ordinary differential equation techniques and an innovatory integral equation approach is employed.

AnH1-Galerkin Expanded Mixed Finite Element Approximation of Second-Order Nonlinear Hyperbolic Equations

We investigate anH1-Galerkin expanded mixed finite element approximation of nonlinear second-order hyperbolic equations, which model a wide variety of phenomena that involve wave motion or convective transport process. This method possesses some features such as approximating the unknown scalar, its gradient, and the flux function simultaneously, the finite element space being free of LBB condition, and avoiding the difficulties arising from calculating the inverse of coefficient tensor. The existence and uniqueness of the numerical solution are discussed. Optimal-order error estimates for this method are proved without introducing curl operator. A numerical example is also given to illustrate the theoretical findings.

Numerical methods for nonlinear second-order hyperbolic partial differential equations. II – Rothe’s techniques for 1-D problems

Two families of Rothe’s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second-order hyperbolic equations with damping and nonlinear source terms are presented. The first family is based on time integration and results in a Volterra integro-differential equation which, upon approximating certain time integrals, can be written as an ordinary differential equation in space. Upon discretizing the spatial derivatives by means of finite difference formula, this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step, and its accuracy is shown to be lower than that of an explicit second-order accurate discretization in both space and time of second-order hyperbolic equations. The second family of methods is based on the discretization of the time derivatives and time linearization, and results in linear ordinary differential equations in space. Upon freezing the coefficients of these equations, one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space, and three-point finite difference equations which depend exponentially on the time step, spatial grid size, and the diffusion, relaxation, damping and reaction times. The finite difference equations are shown to result in non-diagonally dominant matrices unless the time step is smaller than the characteristic relaxation, diffusion, damping and reaction times. To avoid this problem, two Rothe’s techniques that do not account for the Jacobian of the reaction terms in the ordinary differential operator are developed, and it is shown that the Rothe’s techniques belonging to the second family are as accurate as the linearly implicit methods presented in Part I provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space. When the relaxation time is greater than the critical one, Rothe’s methods have been found to be less accurate than the finite difference techniques of Part I. The Rothe’s techniques belonging to the second family are generalized to systems of nonlinear second-order hyperbolic equations and mixed systems of parabolic and second-order hyperbolic partial differential equations.

On the application of mixed finite element method for a strongly nonlinear second-order hyperbolic equation

Mixed finite element method is developed to approximate the solution of the initial-boundary value problem for a strongly nonlinear second-order hyperbolic equation in divergence form. Existence and uniqueness of the approximation are proved, and optimal-orderL∞-in-time,L2-in-space a priori error estimates are derived for both the scalar and vector functions approximated by the method.

Supplement to Cosine Methods for Nonlinear Second-Order Hyperbolic Equations

(2. 1 1) k1(q2Ln1 2Unw-2P2Ln 22n2q Ln L I2wn-) -k {q2[Ln 2w 2wPL(-tnl)(L(tnl )u f(tn))] -k2[p2L n2Un-PL( t)(L(tn)unwfw(Ltn) t )] + q 2 [L n-I 2 U 1-PL.( t n Xd)(L_( t,_ Un t) _f (t n d) ) +kiP(q 2u(4)n 1w2p2u(i)n+q2u(4)n-1w) + 2 k q2P(L(')(tn-l)u(')n 1-2L(t)(tn)u( 1)n'L(')(tn du(1)n1 +2(q -1/1 2)k'PL')(tn)u(I)n +k q2P(L(2)(tr,,, Un 1_2L(2)( tn)un+L(2)(tr, 1) un-1) +(q 1-1/1 2)k'PL(2)(tn ) un _kiq2(f(2)n+1-2f(2)n+f(2)n-l)-k4(q -1/12)f(2)n.

Cosine methods for nonlinear second-order hyperbolic equations

We construct and analyze efficient, high-order accurate methods for approximating the smooth solutions of a class of nonlinear, second-order hyperbolic equations. The methods are based on Galerkin type discretizations in space and on a class of fourth-order accurate two-step schemes in time generated by rational approximations to the cosine. Extrapolation from previous values in the coefficients of the nonlinear terms and use of preconditioned iterative techniques yield schemes whose implementation requires solving a number of linear systems at each time step with the same operator. L 2 {L^2} optimal-order error estimates are proved.

Higher-order single-step fully discrete approximations for nonlinear second-order hyperbolic equations

Error estimates are proved for finite-element approximations to the solution of an initial boundary value problem for nonlinear second-order hyperbolic partial differential equations. Optimal order rates of convergence in L 2 are shown for single-step fully discrete schemes which are linearized by extrapolations. The order of accuracy ν in time is 2, 3 or 4. A class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained.