A new spline in compression method of order four in space and two in time based on halfstep grid points for the solution of the system of 1D quasilinear hyperbolic partial differential equations
 RK Mohanty^{1}Email author and
 Gunjan Khurana^{1, 2}
https://doi.org/10.1186/s1366201711479
© The Author(s) 2017
Received: 23 January 2017
Accepted: 21 March 2017
Published: 29 March 2017
Abstract
In this paper, we propose a new threelevel implicit method based on a halfstep spline in compression method of order two in time and order four in space for the solution of onespace dimensional quasilinear hyperbolic partial differential equation of the form \(u_{tt} =A(x,t,u)u_{xx} +f(x,t,u,u_{x},u_{t})\). We describe spline in compression approximations and their properties using two halfstep grid points. The new method for onedimensional quasilinear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function \(u(x,t)\) and two halfstep points for the known variable ‘x’ in xdirection. The proposed method, when applied to a linear test equation, is shown to be unconditionally stable. We have also established the stability condition to solve a linear fourthorder hyperbolic partial differential equation. Our method is directly applicable to solve hyperbolic equations irrespective of the coordinate system, which is the main advantage of our work. The proposed method for a scalar equation is extended to solve the system of quasilinear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.
Keywords
MSC
1 Introduction
A wave is a time evolution phenomenon that we generally model mathematically using partial differential equations (pdes) which have a dependent variable \(u(x, t)\), which represents the wave value, an independent variable, time t and one or more independent spatial variables. The actual form that the wave takes is strongly dependent upon the system’s initial conditions, boundary conditions and disturbances in the system.
Wave equation is an important secondorder linear partial differential equation for the description of waves as they occur in real life such as ripples on a lake, wind waves on water, tidal surges in estuaries, transverse waves travelling on a long string, transverse vibrations of strings and membranes, traffic density waves, seismic waves, acoustic waves and electromagnetic wave currents in coaxial cables.
Problems involving the propagation of nonlinear waves have become of increasing interest in various branches of science and engineering. In general, waves of finite amplitude governed by a nonlinear evolution equation are called nonlinear waves. As is well known, the principle of superposition of solutions is not valid in nonlinear equations. Therefore the methods familiar to physicists and engineers, like the use of Fourier or Laplace transforms, are no longer applicable with the result that the study of nonlinear waves has not yet become well established. However, in recent years, a number of interesting phenomena involving nonlinear waves have been found, and with the development of digital computers remarkable progress has been made in the research into nonlinear waves.
There has been a consistent effort in developing efficient and high accuracy finite difference methods to solve quasilinear hyperbolic equations. In 1968 to 1969, Bickley and Fyfe [2, 3] developed a cubic spline method for twopoint boundary value problems. Papamichael and Whiteman [4] also developed a cubic spline technique for the solution of onedimensional heat conduction equation. Raggett and Wilson [5] used a cubic spline technique to give a fully implicit finite difference approximation to the onedimensional wave equation. Fleck Jr. [6] proposed a cubic spline method for solving a wave equation of nonlinear optics. Jain and Aziz [7, 8] studied spline function approximations and a cubic spline solution of twopoint boundary value problems with significant first derivative terms. Jain et al. [9] discussed difference schemes based on splines in compression for the solution of conservation laws. Kadalbajoo and Patidar [10, 11] analyzed numerical methods of singularly perturbed twopoint boundary value problems by spline in compression and tension approximations. Khan and Aziz [12] derived a parametric cubic spline approach to the solution of system of twopoint boundary value problems. Kadalbajoo and Aggarwal [13] discussed a cubic spline method for solving singular twopoint boundary value problems. Mohanty et al. [14–17] gave spline in compression methods for singularly perturbed twopoint singular boundary value problems and gave convergent spline in tension methods for singularly perturbed twopoint singular boundary value problems. Rashidinea et al. [18, 19] discussed spline methods for the solution of hyperbolic and parabolic equations. Islam et al. [20, 21] studied nonpolynomial spline approximations for the solution of boundary value problems. Ding and Zhang [22] studied parametric spline methods for the solution of hyperbolic equations. Mohanty and Jain [23] studied the use of a cubic spline method for the solution of 1D quasilinear parabolic equations. Recently, Mohanty et al. [24, 25] derived numerical methods based on nonpolynomial spline approximations for the solution of 1D quasilinear hyperbolic equations. In these methods, they have used fullstep grid points, hence these methods are not directly applicable to problems in polar coordinates. Mohanty et al. [26–35] have also used different techniques for the solution of onedimensional nonlinear wave equations. Most recently, Mohanty and Khurana [36] have proposed a high accuracy numerical method based on offstep discretization for the solution of 2D quasilinear hyperbolic equations. To the authors’ knowledge, no numerical method based on halfstep spline in compression approximation has been developed for the onedimensional quasilinear hyperbolic equation from the consistency condition so far. In this paper, we propose a method derived from the consistency condition, which is applicable to hyperbolic equations irrespective of coordinate systems.
Our paper is arranged as follows. In Section 2, we discuss the properties of spline in compression approximations. In Section 3, we discuss a detailed derivation of a new halfstep threelevel implicit method based on spline in compression approximations. In Section 4, we extend our technique to solve the system of nonlinear secondorder quasilinear hyperbolic equations. In Section 5, we discuss the stability analysis when the method is applied to a telegraphic equation, and we show it to be unconditionally stable. We also establish the stability condition to solve fourthorder linear hyperbolic partial differential equation. In Section 6, we solve some benchmark problems and compare our results with other existing methods. In Section 7, we give concluding remarks.
2 Spline in compression approximations
We discretize the solution domain \([0, 1] \times[0,J]\) into \((N + 1) \times J\) by a set of grid points \((x_{l}, t_{j})\), where \(0 = x_{0} < x_{1} < \cdots< x_{N + 1} = 1\), and \(0 = t_{0} < t_{1} < \cdots< t_{J} = J\), N being a positive integer with uniform mesh spacing \(h = x_{l}  x_{l  1}\), \(k = t_{j}  t_{j  1}\); \(l = 1(1)N + 1\), \(j = 1(1)J\). Let \(u_{l}^{j}\) and \(U_{l}^{j}\) be the approximate and exact solutions of \(u(x, t)\) at the grid point \((x_{l}, t_{j})\), respectively.
3 Method based on nonpolynomial spline in compression approximations
Substituting the above approximations (3.36a) and (3.36b) into (3.12), the order of method (3.12) is retained, and hence we obtain the required numerical method of \(O(k^{2} + k^{2}h^{2} + h^{4})\) based on spline in compression approximations (see [37–45]) for differential equation (1.1).
Note that the initial and Dirichlet boundary conditions are given by (1.2) and (1.3), respectively. Incorporating the initial and boundary conditions, we can write the spline in compression method in a tridiagonal form. If differential equation (1.1) is linear, we use the Gauss elimination (tridiagonal solver) method; in the nonlinear or quasilinear case, we can use the NewtonRaphson iterative method (see [46–48]).
4 Method extended to a system of quasilinear hyperbolic equations
5 Application to a telegraphic equation and stability analysis
In this section we first discuss the background of ‘telegraphic equation’, application of the proposed method to the telegraphic equation with forcing function say f and stability analysis.
It would be difficult to imagine a world without communication systems. A plethora of guided fixed line telephones as well as a multitude of unguided systems to serve cellular phones are evident in our surrounding world. In order to optimize guided communication systems, it is necessary to determine or project power and signal losses in the system since all systems have such losses. To determine these losses and eventually ensure a maximum output, it is necessary to formulate some kind of equation with which to calculate these losses. We give mathematical derivation for the telegraphic equation in terms of voltage and current for a section of a transmission line. The telegraphic equations are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who developed the transmission line model in the 1880s. The theory applies to highfrequency transmission lines (such as telegraph wires and radio frequency conductors), but it is also important for designing highvoltage energy transmission lines. In order to be able to model the telegraphic equations, it is necessary to understand the basic principles of electricity. Ohm’s law describes the relationship between voltage, current and resistance in an electrical circuit. Ohm’s law states that if one volt is applied to one ohm resistance, the current that flows will be one ampere.

\(V= \mbox{voltage measured in volts}\),

\(I= \mbox{current measured in ampere}\),

\(R = \mbox{resistance measured in ohm}\).
The challenge is to model an infinite small piece of telegraph wire as an electric circuit since it has a load and a source as indicated by Ohm’s and Kirchhoff’s laws. The characteristics of a small piece of telegraph wire and that of a long transmission line are the same, thus it is sufficient to model an infinite small piece of telegraph wire to represent a transmission line over distance.

\(x= \mbox{distance from sending end of the cable}\),

\(e(x, t)= \mbox{potential at any point on the cable at any time}\),

\(i(x, t)= \mbox{current at any point on the cable at any time}\),

\(R= \mbox{resistance of the cable}\),

\(L= \mbox{inductance of the cable}\),

\(G= \mbox{conductance to ground}\),

\(C = \mbox{capacitance to ground}\).
Two equations (5.12) and (5.13) are known as the telegraphic equations.
We denote \(a_{0} = \frac{(\alpha_{0} + \beta_{0})^{2}k^{2}}{4}\), \(b_{0} = \alpha_{0}\beta_{0}k^{2}\), \(\lambda= \frac{k}{h}\) and \(f_{l}^{j} = f(x_{l}, t_{j})\).
The additional terms are of high order and do not affect the accuracy of the scheme.
Hence, for \(\alpha_{0} > 0\), \(\beta_{0} \ge0\), \(\eta\ge\frac{1}{64}\), \(\gamma \ge \frac{1 + 3\lambda^{2}}{12\lambda^{2}}\), scheme (5.16) is stable for all choices of \(h > 0\) and \(k > 0\).
6 Numerical results
In this section, we have computed some benchmark problems using the proposed scheme and compared our results obtained by the existing methods for the solution of 1D quasilinear wave equation. The exact solutions are provided in each case. The righthand side homogeneous functions, initial and boundary conditions may be obtained by using the exact solution as a test procedure. The linear difference equations have been solved using tridiagonal solver, whereas nonlinear difference equations have been solved using the NewtonRaphson method. While using the NewtonRaphson method, the iterations were stopped when absolute error tolerance ≤10^{−12} had been achieved. All computations were carried out using MATLAB codes.
The proposed scheme is a threelevel scheme. The value ofuat\(t =0\) is known from the initial condition. To begin the computation, we need the numerical value of u of required accuracy at \(t = k\), so we discuss an explicit method of \(O(k^{2})\) for calculating the value of u at first time level in order to solve the differential equation (1.1) using the proposed scheme (3.12) which is applicable to problems both in Cartesian and polar coordinates.
Order of convergence
Problem no.  Parameters and time  Order 

\(\alpha_{0} = 12\), \(\beta_{0}=8\), η = γ = 1, t = 1, σ = 3.2  3.9834  
\(\alpha_{0} = \beta_{0} = \pi\), η = 0.75, γ = 1.5, t = 1, σ = 3.2  4.0000  
\(\alpha_{0} = 3\pi\), \(\beta_{0} = \pi\), η = 2.5, γ = 0.25, t = 1, σ = 3.2  3.9932  
σ = 0.8, ε = 0.01, t = 1  4.0017  
σ = 0.8, ε = 0.01, t = 2  4.0017  
σ = 0.8, ε = 0.001, t = 1  4.0017  
σ = 0.8, ε = 0.001, t = 2  4.0016  
γ = 0.5, t = 1, σ = 3.2  3.9967  
γ = 2, t = 1, σ = 3.2  3.9944  
γ = 2, t = 1, σ = 3.2  4.0429  
γ = 20, t = 1, σ = 3.2  3.9492  
α = 0.5, t = 1, σ = 1.6  3.9399  
α = 0.05, t = 1, σ = 1.6  3.9987 
Problem 6.1
Telegraphic equation
Problem 6.1 : The maximum absolute errors
h  Proposed method ( 5.16 )  Method discussed in [ 33 ]  

\(\boldsymbol{\alpha_{0} = 12}\) , \(\boldsymbol{\beta_{0} = 8}\) , η = γ = 1  \(\boldsymbol{\alpha_{0} = \beta_{0} = \pi}\) , η = 0.75, γ = 1.5  \(\boldsymbol{\alpha_{0} = 3\pi}\) , \(\boldsymbol{\beta_{0} = \pi}\) , η = 2.5, γ = 0.25  \(\boldsymbol{\alpha_{0} = 12}\) , \(\boldsymbol{\beta_{0} = 8}\) , η = γ = 1  \(\boldsymbol{\alpha_{0} = \beta_{0} = \pi}\) , η = 0.75, γ = 1.5  \(\boldsymbol{\alpha_{0} = 3\pi}\) , \(\boldsymbol{\beta_{0} = \pi}\) , η = 2.5, γ = 0.25  
1/16  7.4557(−07)  2.0167(−06)  1.6177(−06)  5.6408(−06)  4.4195(−06)  1.1784(−06) 
1/32  5.7283(−08)  1.2715(−07)  1.0973(−07)  6.6551(−07)  5.4996(−07)  1.5953(−07) 
1/64  3.6216(−09)  7.9466(−09)  6.8903(−09)  7.2547(−08)  6.4606(−08)  2.0470(−08) 
Problem 6.2
VanderPol type nonlinear wave equation
Problem 6.2 : The maximum absolute errors
h  Proposed method ( 3.12 )  Method discussed in [ 28 ]  

ε = 0.01  ε = 0.001  ε = 0.01  ε = 0.001  
t = 1  t = 2  t = 1  t = 2  t = 1  t = 2  t = 1  t = 2  
1/8  0.4935(−4)  0.3139(−4)  0.4843(−4)  0.3063(−4)  0.1381(−3)  0.8668(−4)  0.1385(−3)  0.8748(−4) 
1/16  0.3068(−5)  0.1951(−5)  0.3011(−5)  0.1904(−5)  0.8596(−5)  0.5395(−5)  0.8620(−5)  0.5445(−5) 
1/32  0.1915(−6)  0.1218(−6)  0.1879(−6)  0.1189(−6)  0.5367(−6)  0.3368(−6)  0.5382(−6)  0.3399(−6) 
Problem 6.3
Nonlinear wave equation
Problem 6.3 : The maximum absolute errors
Problem 6.4
Quasilinear equation
Problem 6.4 : The maximum absolute errors
Problem 6.5
Fourthorder nonlinear hyperbolic equation
Problem 6.5 : The maximum absolute errors
7 Concluding remarks
In this paper, using two halfstep points and a central point, we have derived a new stable halfstep spline in compression method of \(O(k^{2} + h^{4})\) accuracy for the solution of quasilinear hyperbolic equation (1.1). Our method has been derived directly from the consistency condition which is fourthorder accurate, and we have used properties of spline in compression function in derivation of the method. For a fixed parameter \(\sigma= k/h^{2}\), the proposed method behaves like a fourthorder method. The accuracy and efficiency of the proposed method are exhibited from the numerical computations. The proposed method for scalar equation has been extended in a vector form to solve the system of quasilinear hyperbolic pdes. For the telegraphic equation, the method is shown to be unconditionally stable, and the stability condition for solving a fourthorder linear hyperbolic pde has also been established. The method is directly applicable to quasilinear hyperbolic pdes irrespective of the coordinate system, which brings an edge over other existing methods.
Declarations
Acknowledgements
This work is supported by I.P. College for Women, University of Delhi. The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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