Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.Comment: submitted to Siam SIA

A semi analytic iterative method for solving two forms of blasius equation / Mat Salim Selamat, Nurul Atkah Halmi and Nur Azyyati Ayob

In this paper, a semi analytic iterative method (SAIM) is presented for solving two forms of Blasius equation. Blasius equation is a third order nonlinear ordinary differential equation in the problem of the two-dimensional laminar viscous flow over half-infinite domain. In this scheme, the first solution which is in a form of convergent series solution is combined with Padé approximants to handle the boundary condition at infinity. Comparison the results obtained by SAIM with those obtained by other method such as variational iteration method and differential transform method revealed the effectiveness of the SAIM

The instanton method and its numerical implementation in fluid mechanics

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations

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

A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.This work focuses on the Parareal parallelin-time method and its application to the viscous Burgers equation. A crucial component of Parareal is the coarse time stepping scheme, which strongly impacts the convergence of the parallel-in-time method. Three choices of coarse time stepping schemes are investigated in this work: explicit Runge-Kutta, implicit-explicit Runge-Kutta, and implicit Runge-Kutta with semiLagrangian advection. Manufactured solutions are used to conduct studies, which provide insight into the viability of each considered time stepping method for the coarse time step of Parareal. One of our main findings is the advantageous convergence behavior of the semi-Lagrangian scheme for advective flows.Schmitt: The work of this author is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universit¨at Darmstadt Peixoto: Acknowledges the Sao Paulo Research Foundation (FAPESP) under the grant number 2016/18445-7 and the National Science and Technology Development Council (CNPq) under grant number 441328/2014-