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Free Fall and Harmonic Oscillators
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
The problem with the solution is that Euler’s Method is not an energy conserving method. As conservation of energy is important in physics, we would like to be able to seek problems that conserve energy. Such schemes used to solve oscillatory problems in classical mechanics are called symplectic integrators. A simple example is the Euler–Cromer, or semi-implicit Euler Method. We only need to make a small modification of Euler’s Method. Namely, in the second equation of the method, we use the updated value of the dependent variable as computed in the first line.
An improved stiff-ODE solving framework for reacting flow simulations with detailed chemistry in OpenFOAM
Published in Combustion Theory and Modelling, 2023
Kun Wu, Yuting Jiang, Zhijie Huo, Di Cheng, Xuejun Fan
As the BDF method is introduced by coupling with external CVODE package, hence the computational efficiency comparison between Radau-IIA and BDF methods is not straightforward. Hence, the Seulex method shipped with the OpenFOAM standard ODE solver library is taken as the benchmark mainly for efficiency assessment. The Seulex method is an extrapolation-based algorithm that uses a sequence of lower-order solutions to project high-order approximation of . This method splits the integration step into several sub-intervals , where is defined by a sequence [17]: (i.e.: 2, 3, 4, 6, 8, 12, …). Afterward, in each sub-interval, a first-order semi-implicit Euler method is used to solve Equation (3) using the sub-interval size . The low-order solution is then used to successively build higher-order approximations via the Aitken-Neville Algorithm [15,17]. As Imren and Haworth suggested that the Seulex method offers significant advantages in accuracy and computational efficiency compared to the BDF method [15], it has become the most preferred stiff ODE solver in the OpenFOAM platform nowadays [36].
Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity
Published in Inverse Problems in Science and Engineering, 2021
Everton Boos, Vanda M. Luchesi, Fermín S. V. Bazán
As mentioned above, mathematically equivalent applications for the conduction problem include reconstruction of electrical conductivity. For instance, in medical imaging applications, the numerical reconstructions of anisotropic conductivity arises in electrical impedance tomography [2–4,22]. Several techniques for computing anisotropic conductivity in this context have been used. In particular, in thermal tomography [23], where the coefficient K is characterized by a finite number of parameters in a high-dimensional subset of referred to as parameter domain, the direct problem is solved by coupling the finite element methods (FEM) in the spatial domain and a semi-implicit Euler method in the time interval. On its turn, the inverse problem is formulated as a non-linear least squares problem coupled with Tikhonov regularization. In contrast, in [24], the parameter identification problem is modelled as a variational problem over stochastic Sobolev spaces, where a spectral approximation of the observation field is used to estimate the solution problem using a finite noisy model. In resume, most of these techniques approach the partial differential equation given by (1), with Neumann, Dirichlet or mixed boundary conditions, by using one or more numerical methods, then an objective function is formulated and an optimization technique is chosen to find an optimal parameter.
Numerical Modeling of a Wicked Heat Pipe Using Lumped Parameter Network Incorporating the Marangoni Effect
Published in Heat Transfer Engineering, 2021
Jibin Joy Kolliyil, Naresh Yarramsetty, Chakravarthy Balaji
A parametric study was conducted with the use of the lumped parameter model. The dimensions and other geometric characteristics of heat pipe were taken to be the same as that of the heat pipe used by Huang et al. [7] and reported by Tournier and El-Genk [8] for their experiments. All the wicks were assumed to be made of screen mesh copper wick. The semi-implicit Euler method was used to solve the system of equations for faster convergence. The following parameters were changed - the heat input, wick thickness, effective wick porosity, external pipe diameter and the working fluid. The thermal resistance is the key figure of merit in a heat pipe and is given by