IMPLEMENTASI PENYELESAIAN PERSAMAAN BURGERS DENGAN METODE BEDA HINGGA DALAM BAHASA PEMROGRAMAN JULIA

Burgers equation is a partial differential equation used to modelling several events related to fluids. Burgers equation was firstly introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. This study discusses solving Burgers equations with finite difference method. In this study, several parameters have been known for the Burgers equation and several cases of partitions are used in finite difference method. The result shows that the more partitions used, the numerical result obtained will be closer to the exact values. In this study, calculations are numerically carried out with the help of Julia programming language

(SI10-067) Numerical Study of the Time Fractional Burgers’ Equation by Using Explicit and Implicit Schemes

The study discusses the numerical solution for a time fractional Burgers’ equation using explicit (scheme 1) and implicit scheme (scheme 2), respectively. The approximation of the differential equation is discretized using the finite difference method (FDM). A non-linear term present in the Burgers’ equation is approximated using the time-averaged values. The Von-Neumann analysis shows that the Scheme 1 is conditionally stable and Scheme 2 is unconditionally stable. The numerical solutions are compared with the exact solutions and are good in agreement. Also, the error is estimated between exact and numerical solutions

Proper orthogonal decomposition closure models for fluid flows: Burgers equation

This paper puts forth several closure models for the proper orthogonal decomposition (POD) reduced order modeling of fluid flows. These new closure models, together with other standard closure models, are investigated in the numerical simulation of the Burgers equation. This simplified setting represents just the first step in the investigation of the new closure models. It allows a thorough assessment of the performance of the new models, including a parameter sensitivity study. Two challenging test problems displaying moving shock waves are chosen in the numerical investigation. The closure models and a standard Galerkin POD reduced order model are benchmarked against the fine resolution numerical simulation. Both numerical accuracy and computational efficiency are used to assess the performance of the models

On the Cauchy problem of dispersive Burgers type equations

We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+\partial_x [T_u u]-T_{\frac{\partial_x u}{2}}u+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ which contains the main non linear "worst interaction" terms, i.e low-high interaction terms, of the usual weakly dispersive Burgers type equation: $$\partial_t u+u\partial_x u+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[,$$ with $u_0 \in H^s(\mathbb D)$, where $\mathbb D=\mathbb T \text{ or } \mathbb R$. Through a paradifferential complex Cole-Hopf type gauge transform we introduce for the study of the flow map regularity of Gravity-Capillary equation, we prove a new a priori estimate in $H^s(\mathbb D)$ under the control of $\left\Vert(1+\left\Vert u\right\Vert_{L^\infty_x})\left\Vert u \right\Vert_{W^{2-\alpha,\infty}_x}\right\Vert_{L^1_t}$, improving upon the usual hyperbolic control $\left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x}$. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured by J. C. Saut and C. Klein in their numerical study of the dispersive Burgers equation. For $\alpha\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_tu+ \partial_x |D|^{\alpha-1}u=R_\infty(u),\ \alpha \in ]2,3[,$$ where $R_\infty$ is a regularizing operator under the control of $\left\Vert u\right\Vert_{L^\infty_t C^{2-\alpha}_*}$

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