Dimensional Hyperbolic Equation Research Articles

Lagrangian formulation of a generalised coupled hyperbolic system

We perform Noether classification of the generalised system of coupled (2 + 1)-dimensional hyperbolic equations, namely utt-uxx-uyy+P(v)=0,vtt-vxx-vyy+Q(u)=0. Besides this we compute conservation laws corresponding to cases that have Noether symmetries for the underlying coupled hyperbolic system.

Numerical solution of second order one dimensional hyperbolic equation by exponential B-spline collocation method

Numerical solution of second order one dimensional hyperbolic equation by exponential B-spline collocation method

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

Abstract We first introduced a linear stationary equation with a quadratic operator in ∂ x and ∂ y , then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients

Purpose – The purpose of this paper is to present the computational modeling of second-order two-dimensional nonlinear hyperbolic equations by using cosine expansion-based differential quadrature method (CDQM). Design/methodology/approach – The CDQM reduced the equations into a system of second-order differential equations. The obtained system is solved by RK4 method by converting into a system of first ordinary differential equations. Findings – The computed numerical results are compared with the results presented by other workers (Mohanty et al., 1996; Mohanty, 2004) and it is found that the present numerical technique gives better results than the others. Second, the proposed algorithm gives good accuracy by using very less grid point and less computation cost as comparison to other numerical methods such as finite difference methods, finite elements methods, etc. Originality/value – The author extends CDQM proposed in (Korkmaz and Dağ, 2009b) for two-dimensional nonlinear hyperbolic partial differential equations. This work is new for two-dimensional nonlinear hyperbolic partial differential equations.

New explicit group iterative methods in the solution of two dimensional hyperbolic equations

In this paper, we present the development of new explicit group relaxation methods which solve the two dimensional second order hyperbolic telegraph equation subject to specific initial and Dirichlet boundary conditions. The explicit group methods use small fixed group formulations derived from a combination of the rotated five-point finite difference approximation together with the centered five-point centered difference approximation on different grid spacings. The resulting schemes involve three levels finite difference approximations with second order accuracies. Analyses are presented to confirm the unconditional stability of the difference schemes. Numerical experimentations are also conducted to compare the new methods with some existing schemes.

Stable difference schemes for the hyperbolic problems subject to nonlocal boundary conditions with self-adjoint operator

Abstract In the present paper the first and second orders of accuracy difference schemes for the numerical solution of multidimensional hyperbolic equations with nonlocal boundary and Dirichlet conditions are presented. The stability estimates for the solution of difference schemes are obtained. A method is used for solving these difference schemes in the case of one dimensional hyperbolic equation.

ON MULTIPOINT NONLOCAL BOUNDARY VALUE PROBLEMS FOR HYPERBOLIC DIFFERENTIAL AND DIFFERENCE EQUATIONS

The nonlocal boundary value problem for differential equation \[\left\{ \begin{array}{l} \frac{d^{2}u(t)}{dt^{2}} + Au(t) = f(t) \quad \ (0 \leq t \leq 1), \\ u(0) = \sum\limits_{r=1}^{n} \alpha _{r} u(\lambda_{r}) + \varphi, u_{t}(0) = \sum\limits_{r=1}^{n} \beta _{r} u_{t}(\lambda_{r}) + \psi, \\ 0 \lt \lambda_{1} \leq \lambda_{2} \leq ... \leq \lambda_{n} \leq 1 \end{array} \right. \] in a Hilbert space $H$ with the self-adjoint positive definite operator $A$ is considered. The stability estimates for the solution of the problem under the assumption \[ \sum_{k=1}^{n} \left\vert \alpha_{k} + \beta_{k} \right\vert + \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert \sum_{\substack{m=1 \\ m \neq k}}^{n} \left\vert \beta_{m} \right\vert \lt |1 + \sum_{k=1}^{n} \alpha_{k} \beta_{k}| \] are established. The first order of accuracy difference schemes for the approximate solutions of the problem are presented. The stability estimates for the solution of these difference schemes under the assumption \[ \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert + \sum_{k=1}^{n} \left\vert \beta_{k} \right\vert + \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert \sum_{k=1}^{n} \left\vert \beta_{k} \right\vert \lt 1 \] are established. In practice, the nonlocal boundary value problems for one dimensional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional hyperbolic equation with Dirichlet condition in space variables are considered. The stability estimates for the solutions of difference schemes for the nonlocal boundary value hyperbolic problems are obtained.

An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients

We propose a three level implicit unconditionally stable difference scheme of O( k 2 + h 2) for the difference solution of second order linear hyperbolic equation u tt + 2 α( x, t) u t + β 2( x, t) u = A( x, t) u xx + f( x, t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where A( x, t) > 0, α( x, t) > β( x, t) ⩾ 0. The proposed formula is applicable to the problems having singularity at x = 0. The resulting tri-diagonal linear system of equations is solved by using Gauss-elimination method. Numerical examples are provided to illustrate the unconditionally stable character of the proposed method.

An operator splitting technique for an unconditionally stable difference method for a linear three space dimensional hyperbolic equation with variable coefficients

We report a new three-step operator splitting method of O( k 2+ h 2) for the difference solution of linear hyperbolic equation u tt +2 α( x, y, z, t) u t + β 2( x, y, z, t) u= A( x, y, z, t) u xx + B( x, y, z, t) u yy + C( x, y, z, t) u zz + f( x, y, z, t) subject to appropriate initial and Dirichlet boundary conditions, where α( x, y, z, t)> β( x, y, z, t)>0 and A( x, y, z, t)>0, B( x, y, z, t)>0, C( x, y, z, t)>0. The method is applicable to singular problems and stable for all choices of h>0 and k>0. The resulting system of algebraic equations is solved by using a tri-diagonal solver. Computational results are provided to demonstrate the viability of the new method.

Linear stability analysis and fourth‐order approximations at first time level for the two space dimensional mildly quasi‐linear hyperbolic equations

AbstractIn 1996, Mohanty et al. [1] presented a fourth‐order finite difference solution of a two space dimensional nonlinear hyperbolic equation with Dirichlet boundary conditions. In 1998, Mohanty et al. [2] discussed a fourth‐order approximation at first time level for the numerical solution of the one space dimensional hyperbolic equation. In both the cases, they have discussed the stability analysis for the linear hyperbolic equation having first‐order space derivative terms. Recently, Mohanty et al. [3] have developed fourth‐order difference formulas for the three space dimensional quasi‐linear hyperbolic equations and obtained fourth‐order approximation at first time level. In this article, we extend our strategy for solving the two space dimensional quasi‐linear hyperbolic equation. An operator splitting method for a linear hyperbolic equation having a time derivative term is proposed. Linear stability analysis and fourth‐order approximation at first time level for the two space dimensional quasi‐linear hyperbolic equation are also discussed. The results of the numerical experiments are compared with the exact solution. © 2001 John Wiley &amp; Sons, Inc. Numer Methods Partial Differential Eq 17: 607–618, 2001

Numerically Nonreflecting Boundary and Interface Conditions for Compressible Flow and Aeroacoustic Computations

Accuratenonreecting orradiationboundaryconditionsareimportantforeffectivecomputation ofaeroacoustic and compressibleow problems. Theperformance of such boundary conditions is often degraded upon discretiza- tion of the equations with ® nite difference and time marching methods. In particular, poorly resolved, spurious sawtooth waves are generated at boundaries due to the dispersive nature of the ® nite difference approximation. These disturbances can lead to spurious self-sustained oscillations in theow (self-forcing), poor convergence to steady state, and long timeinstability ofthenumerics. Exact discretely nonreecting boundary closures (boundary conditionsforadownwindarti® cialboundary andanupwindphysicalboundary)arederived by considering aone- dimensional hyperbolic equation discretized with ® nite difference schemes and Runge± Kutta time advancements. The current methodology leads to stable local ® nite difference-like boundary closures, which are nonreecting to an essentially arbitrarily high order of accuracy. These conditions can also be applied at interfaces where there is a discontinuity in the wave speed (a shock) or where there is an abrupt change in the grid spacing. Compared to other boundary treatments, the present boundary and interface conditions can reduce spurious reected energy in the computational domain by many orders of magnitude.

High accuracy difference schemes for a class of singular three space dimensional hyperbolic equations

For the numerical integration of the system of 3-D nonlinear hyperbolic equations with variable coefficients, we report two three-level implicit difference methods of 0(k 4 + k 2 h 2 + h 4) where k and h are grid sizes in time and space directions, respectively. When the coefficients of uxy, uyz and uyzare equal to zero we require only (7+19 + 7) grid points and when the coefficients of uxy, uyz and uzx are not equal to zero and the coefficients of uxx, uyy and uzz are equal we require (19+27+19) grid points. The three-level conditionally stable ADI method of 0 (k4 + k 2 h 2+ h 4) for the numerical solution of wave equation in polar coordinates is discussed. Numerical examples are provided to illustrate the methods and their fourth order convergence.

Third and fourth order accuracy schemes for two dimensional hyperbolic equations

Abstract : In this paper the Lax-Wendroff procedure is extended to the scalar case of a two-dimensional hyperbolic conservation law. Explicit third and fourth order accuracy finite-difference operators are constructed for solving quasi-linear initial value problems. Stability conditions are obtained and are used in numerical computations. The computational results which are presented demonstrate that large amounts of computing time and memory space are saved without loss of accuracy. (Author)

A Classification of Split Difference Methods for Hyperbolic Equations in Several Space Dimensions

A Classification of Split Difference Methods for Hyperbolic Equations in Several Space Dimensions