Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

[EN] In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Smoothed Particle Interpolation

Smoothed particle hydrodynamics (SPH) discretization techniques are generalized to develop a method, smoothed particle interpolation (SPI), for solving initial value problems of systems of non-hydrodynamical nature. Under this approach, SPH is viewed as strictly an interpolation scheme and, as such, suitable for solving general hyperbolic and parabolic equations. The SPI method is tested on (1) the wave equation with inhomogeneous sound speed and (2) Burgers equation. The efficiency of SPI is studied by comparing SPI solutions to those obtained with standard finite difference methods. It is shown that the power of SPI arises when the smoothing particles are free to move.Comment: 13 pages (LaTeX), 9 figures (not included), [email protected]

A Review on Higher Order Spline Techniques for Solving Burgers Equation using B-Spline methods and Variation of B-Spline Techniques

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers' equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these method

An integral-like numerical approach for solving Burgers' equation

An integral-like approach established on spline polynomial interpolations is applied to the one-dimensional Burgers' equation. The Hopf-Cole transformation that converts non-linear Burgers' equation to linear diffusion problem is emulated by using Taylor series expansion. The diffusion equation is then solved by using analytic integral formulas. Four experiments were performed to examine its accuracy, stability and parallel scalability. The correctness of the numerical solutions is evaluated by comparing with exact solution and assessed error norms. Due to its integral-like characteristic, large time step size can be employed without loss of accuracy and numerical stability. For practical applications, at least cubic interpolation is recommended. Parallel efficiency seen in the weak-scaling experiment depends on time step size but generally adequate.Comment: Under submission. 20 pages, 5 figures, 6 table

Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation

This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modi ed Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation nite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms L2, L1 and three invariants I1; I2 and I3 are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation