A semi-discretization method based on quartic splines for solving one-space-dimensional hyperbolic equations

An Unconditionally Stable Spline Difference Scheme for Solving the Second 2D Linear Hyperbolic Equation

“…The initial, boundary and right-hand side homogeneous functions can be obtained using the exact solution as a test procedure. Note that the proposed difference scheme is implicit three level scheme, to start my computation, it is necessary to know the value of u of required accuracy at all the nodal points at first time level, that is, at t = t 1 = k. Following [12], we may obtain the numerical solution of u at t = k.…”

Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach

“…The boundary conditions given in equations (9) or (10) are used for two additional linear equations to obtain a unique solution of the resulting system. Thus, the system becomes a matrix system of dimension ( 3) ( 3) NN    which is a tri-diagonal system that can be solved by the Thomas Algorithm [29][30][31][32][33].…”

“…The numerical results are found to be in good agreement with known exact solutions and also with earlier studies.Keywords: Second order one dimensional telegraph equation, Cubic trigonometric B-spline basis functions, Cubic trigonometric B-spline collocation method, Stability * Corresponding Email address: m.abbas@uos.edu.pk 2 1.3.Literature review Several numerical methods have been developed to solve the telegraph equation subject to Dirichlet boundary conditions and the references are in [2,[5][6][7][8]. In [9], two semi-discretization methods based on quartic splines function have been developed to solve the telegraph equations. A class of unconditionally stable finite difference schemes constructed with the help of quartic splines functions has been developed by Liu and Liu [10] for the solution of the telegraph equation.…”

High order locally one-dimensional methods for solving two-dimensional parabolic equations

“…All these schemes obtain fourth order accuracy in space, but only second order accuracy in time, since the Crank-Nicolson implicit method is employed for time discretization. Very recently, a semi-discrete method and Padé approximations or the Runge-Kutta methods were exploited to increase the temporal accuracy [19][20][21][22][23][24]. Vu and Alexander [19] developed a series of explicit exponential Runge-Kutta methods of high order for parabolic problems.…”

“…They obtained two schemes with third and fifth order accuracy in time, respectively. Liu et al [21] used a similar strategy, (2,2) and (3,3) Padé approximations for the time discretization and C 3 quartic spline approximation for space discretization, to get two higher order difference schemes for the one-dimensional linear hyperbolic equation. Zhang [23] provided a (3, 3) Padé approximation method for solving space fractional Fokker-Planck equations.…”