scholarly journals Nonstandard finite-difference schemes for the two-level Bloch model

2018 ◽  
Vol 09 (04) ◽  
pp. 1850033 ◽  
Author(s):  
Marc E. Songolo ◽  
Brigitte Bidégaray-Fesquet
Keyword(s):  
Finite Difference ◽  
Physical Properties ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Bloch Equation ◽  
Finite Difference Schemes ◽  
Splitting Schemes ◽  
Matrix Exponentials ◽  
Nonstandard Finite Difference

In this paper, we present splitting schemes for the two-level Bloch model. After proposing two ways to split the Bloch equation, we show that it is possible in each case to generate exact numerical solutions of the obtained sub-equations. These exact solutions involve matrix exponentials which can be expensive to compute. Here, for [Formula: see text] matrices we develop equivalent formulations which reduce the computational cost. These splitting schemes are nonstandard ones and conserve all the physical properties (Hermicity, positiveness and trace) of Bloch equations. In addition, they are explicit, making effective their implementation when coupled with the Maxwell’s equations.

Combinations of nonstandard finite difference schemes and composition methods with complex time steps for population models

2016 ◽  
Vol 09 (04) ◽  
pp. 1650049
Author(s):  
Cuicui Liao ◽  
Xiaohua Ding ◽  
Jiuzhen Liang
Keyword(s):  
Finite Difference ◽  
Numerical Method ◽  
Numerical Solutions ◽  
Whooping Cough ◽  
Population Models ◽  
Finite Difference Schemes ◽  
Complex Time ◽  
Composition Methods ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

We propose an efficient numerical method for two population models, based on the nonstandard finite difference (NSFD) schemes and composition methods with complex time steps. The NSFD scheme is able to give positive numerical solutions that satisfy the conservation law, which is a key property for biological population models. The accuracy is improved by using the composition methods with complex time steps. Numerical tests on the plankton nutrient model and whooping cough model are presented to show the efficiency and advantage of the proposed numerical method.

scholarly journals Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 667
Author(s):  
Maryam Almutairi ◽  
Hamzeh Zureigat ◽  
Ahmad Izani Ismail ◽  
Ali Fareed Jameel
Keyword(s):  
Finite Difference ◽  
Fuzzy Number ◽  
Wave Equations ◽  
Fuzzy Numbers ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Finite Difference Schemes ◽  
Numerical Schemes ◽  
Implicit Schemes ◽  
Number Form

The use of fuzzy partial differential equations has become an important tool in which uncertainty or vagueness exists to model real-life problems. In this article, two numerical techniques based on finite difference schemes that are the centered time center space and implicit schemes to solve fuzzy wave equations were used. The core of the article is to formulate a new form of centered time center space and implicit schemes to obtain numerical solutions fuzzy wave equations in the double parametric fuzzy number approach. Convex normalized triangular fuzzy numbers are represented by fuzziness, based on a double parametric fuzzy number form. The properties of fuzzy set theory are used for the fuzzy analysis and formulation of the proposed numerical schemes followed by the new proof stability thermos under in the double parametric form of fuzzy numbers approach. The consistency and the convergence of the proposed scheme are discussed. Two test examples are carried out to illustrate the feasibility of the numerical schemes and the new results are displayed in the forms of tables and figures where the results show that the schemes have not only been effective for accuracy but also for reducing computational cost.

Acoustic and elastic modeling by optimal time-space-domain staggered-grid finite-difference schemes

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T17-T40 ◽  
Author(s):  
Zhiming Ren ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Computational Cost ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Equation Modeling ◽  
Optimal Time ◽  
Finite Difference Schemes ◽  
Space Domain ◽  
Time Space

Staggered-grid finite-difference (SFD) methods are widely used in modeling seismic-wave propagation, and the coefficients of finite-difference (FD) operators can be estimated by minimizing dispersion errors using Taylor-series expansion (TE) or optimization. We developed novel optimal time-space-domain SFD schemes for acoustic- and elastic-wave-equation modeling. In our schemes, a fourth-order multiextreme value objective function with respect to FD coefficients was involved. To yield the globally optimal solution with low computational cost, we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least-squares (LS) to derive the optimal SFD coefficients by minimizing the relative error of time-space-domain dispersion relations over a given frequency range. To ensure the robustness of our schemes, a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold. Moreover, the hybrid absorbing boundary condition was applied to remove artificial boundary reflections. We compared our optimal SFD with the conventional, TE-based time-space-domain, and LS-based SFD schemes. Dispersion analysis and numerical simulation results suggested that the new SFD schemes had a smaller numerical dispersion than the other three schemes when the same operator lengths were adopted. In addition, our LS-based time-space-domain SFD can obtain the same modeling accuracy with shorter spatial operator lengths. We also derived the stability condition of our schemes. The experiment results revealed that our new LS-based SFD schemes needed a slightly stricter stability condition.

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