scholarly journals Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 206
Author(s):  
María Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Stochastic Process ◽  
Finite Difference ◽  
Coming Out ◽  
Finite Difference Methods ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Finite Degree ◽  
Mean Square Stability ◽  
Reaction Models ◽  
Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

Characteristics of finite difference methods for dispersive shallow water equations

2016 ◽  
Vol 31 (3) ◽  
Author(s):  
Zinaida I. Fedotova ◽  
Gayaz S. Khakimzyanov
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hydrodynamic Equations ◽  
Difference Methods

AbstractThe paper contains a description of the most important properties of numerical methods for solving nonlinear dispersive hydrodynamic equations and their distinctions from similar properties of finite difference schemes approximating classic dispersion-free shallow water equations.

scholarly journals Finite Difference Methods for Weather Prediction in Abuja, Nigeria

2020 ◽  
pp. 915-931
Author(s):  
Jacob Emmanuel ◽  
Ogunfiditimi F.O. ◽  
Victor Alexander Okhuese ◽  
Odeyemi J. K
Keyword(s):  
Finite Difference ◽  
Weather Prediction ◽  
Finite Difference Methods ◽  
Weather Forecast ◽  
Difference Schemes ◽  
Weather Data ◽  
Finite Difference Schemes ◽  
Numerical Weather ◽  
Difference Methods ◽  
The Finite Difference Method

In this research, we have been able to simulate some finite difference schemes to predict weather trends of Abuja Station, Nigeria. By analyzing the results from these schemes, it has shown that the best scheme in the finite difference method that gives a close accurate weather forecast is the trapezoidal scheme hence we use it to simulate numerical weather data obtained from Federal Airports Authority of Nigeria (FAAN), Abuja and corresponding numerical weather data obtained by the compatible finite difference schemes, using MATLAB (R2012a) software to obtain future numerical weather trends.

scholarly journals Simple bespoke preservation of two conservation laws

2018 ◽  
Vol 40 (2) ◽  
pp. 1294-1329 ◽  
Author(s):  
Gianluca Frasca-Caccia ◽  
Peter Ellsworth Hydon
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Vries Equation ◽  
Finite Difference Methods ◽  
Nonlinear Heat Equation ◽  
Finite Difference Schemes ◽  
Simplified Approach ◽  
Numerical Tests ◽  
Difference Methods ◽  
Discrete Conservation Laws

Abstract Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic–numeric approach feasible. To illustrate the simplified approach we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg–de Vries equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared with others in the literature.

Non-standard finite-difference methods for vibro-impact problems

2005 ◽  
Vol 461 (2058) ◽  
pp. 1927-1950 ◽  
Author(s):  
Yves Dumont ◽  
Jean M.-S Lubuma
Keyword(s):  
Finite Difference ◽  
Duffing Oscillator ◽  
Finite Difference Methods ◽  
Finite Difference Schemes ◽  
Restitution Coefficient ◽  
Conservation Of Energy ◽  
Numerical Examples ◽  
Impact Oscillators ◽  
Difference Methods ◽  
Impact Problems

Impact oscillators are non-smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful. We design non-standard finite-difference schemes in which the intrinsic qualitative parameters of the system—the restitution coefficient, the oscillation frequency and the structure of the nonlinear terms—are suitably incorporated. The schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times. Numerical examples, including the Duffing oscillator that develops a chaotic behaviour for some positions of the obstacle, are presented. It is observed that the cpu times of computation are of the same order for both the standard and the non-standard schemes.

A NOTE ON THE LATTICE BOLTZMANN VERSUS FINITE-DIFFERENCE METHODS FOR THE NUMERICAL SOLUTION OF THE FISHER'S EQUATION

2013 ◽  
Vol 25 (01) ◽  
pp. 1340015 ◽  
Author(s):  
SAURO SUCCI
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Fisher’S Equation ◽  
Advection Diffusion ◽  
Fisher's Equation ◽  
Better Than

We assess the Lattice Boltzmann (LB) method versus centered finite-difference schemes for the solution of the advection–diffusion–reaction (ADR) Fisher's equation. It is found that the LB method performs significantly better than centered finite-difference schemes, a property we attribute to the near absence of dispersion errors.

scholarly journals Some Finite Difference Methods to Model Biofilm Growth and Decay: Classical and Non-Standard

Computation ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 123
Author(s):  
Yusuf Olatunji Tijani ◽  
Appanah Rao Appadu ◽  
Adebayo Abiodun Aderogba
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Biofilm Formation ◽  
Defense Mechanism ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Biofilm Growth ◽  
The Mathematical Model

The study of biofilm formation is undoubtedly important due to micro-organisms forming a protected mode from the host defense mechanism, which may result in alteration in the host gene transcription and growth rate. A mathematical model of the nonlinear advection–diffusion–reaction equation has been studied for biofilm formation. In this paper, we present two novel non-standard finite difference schemes to obtain an approximate solution to the mathematical model of biofilm formation. One explicit non-standard finite difference scheme is proposed for biomass density equation and one property-conserving scheme for a coupled substrate–biomass system of equations. The nonlinear term in the mathematical model has been handled efficiently. The proposed schemes maintain dynamical consistency (positivity, boundedness, merging of colonies, biofilm annihilation), which is revealed through experimental observation. In order to verify the accuracy and effectiveness of our proposed schemes, we compare our results with those obtained from standard finite difference schemes and earlier known results in the literature. The proposed schemes (NSFD1 and NSFD2) show good performance. The NSFD2 scheme reveals that the processes of biofilm formation and nutritive substrate growth are intricately linked.

scholarly journals Generating Solar Sail Trajectories in the Earth-Moon System Using Augmented Finite-Difference Methods

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Geoffrey G. Wawrzyniak ◽  
Kathleen C. Howell
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Orbital Plane ◽  
Solar Sail ◽  
Finite Difference Schemes ◽  
Trajectory Design ◽  
Dynamical Regime ◽  
The Earth ◽  
Collocation Scheme ◽  
Difference Methods

Using a solar sail, a spacecraft orbit can be offset from a central body such that the orbital plane is displaced from the gravitational center. Such a trajectory might be desirable for a single-spacecraft relay to support communications with an outpost at the lunar south pole. Although trajectory design within the context of the Earth-Moon restricted problem is advantageous for this problem, it is difficult to envision the design space for offset orbits. Numerical techniques to solve boundary value problems can be employed to understand this challenging dynamical regime. Numerical finite-difference schemes are simple to understand and implement. Two augmented finite-difference methods (FDMs) are developed and compared to a Hermite-Simpson collocation scheme. With 101 evenly spaced nodes, solutions from the FDM are locally accurate to within 1740 km. Other methods, such as collocation, offer more accurate solutions, but these gains are mitigated when solutions resulting from simple models are migrated to higher-fidelity models. The primary purpose of using a simple, lower-fidelity, augmented finite-difference method is to quickly and easily generate accurate trajectories.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

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