A Time Operator Splitting Method for Numerical Analysis of 1-D Solid Propellant Non-Linear Combustion Response

2003 ◽  
Author(s):  
Mehmet Ali Ak ◽  
Huseyin Vural
Keyword(s):  
Numerical Analysis ◽  
Solid Propellant ◽  
Operator Splitting ◽  
Splitting Method ◽  
Time Operator ◽  
Operator Splitting Method ◽  
Non Linear

An efficient space-time operator-splitting method for high-dimensional vector-valued Allen–Cahn equations

2019 ◽  
Vol 29 (9) ◽  
pp. 3437-3453 ◽  
Author(s):  
Yunxia Sun ◽  
Xufeng Xiao ◽  
Zhiming Gao ◽  
Xinlong Feng
Keyword(s):  
Upper Bound ◽  
Operator Splitting ◽  
Splitting Method ◽  
Mass Conservation ◽  
Space Time ◽  
Time Operator ◽  
High Dimensional ◽  
Heat Equations ◽  
Content Type ◽  
Operator Splitting Method

Purpose The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations. Design/methodology/approach Each component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement. Findings The theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method. Practical implications Extensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method. Originality/value The space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.

scholarly journals The Sum and Difference of Two Lognormal Random Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
C. F. Lo
Keyword(s):  
Stochastic Process ◽  
Probability Distributions ◽  
Operator Splitting ◽  
Splitting Method ◽  
Unified Approach ◽  
New Approach ◽  
Numerical Examples ◽  
Operator Splitting Method ◽  
Approximate Probability ◽  
Analytical Series

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum ofNlognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

scholarly journals COSMO: A Conic Operator Splitting Method for Convex Conic Problems

2021 ◽  
Vol 190 (3) ◽  
pp. 779-810
Author(s):  
Michael Garstka ◽  
Mark Cannon ◽  
Paul Goulart
Keyword(s):  
Convex Sets ◽  
Operator Splitting ◽  
Splitting Method ◽  
Computational Cost ◽  
Coefficient Matrix ◽  
Benchmark Problems ◽  
Portfolio Optimisation ◽  
Splitting Algorithm ◽  
Semidefinite Programs ◽  
Operator Splitting Method

AbstractThis paper describes the conic operator splitting method (COSMO) solver, an operator splitting algorithm and associated software package for convex optimisation problems with quadratic objective function and conic constraints. At each step, the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to effectively exploit sparsity in large, structured semidefinite programs. Numerical comparisons with other state-of-the-art solvers for a variety of benchmark problems show the effectiveness of our approach. Our Julia implementation is open source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem.

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