scholarly journals Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

Computation ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 92
Author(s):  
Ádám Nagy ◽  
Issa Omle ◽  
Humam Kareem ◽  
Endre Kovács ◽  
Imre Ferenc Barna ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Initial Conditions ◽  
Numerical Algorithms ◽  
Time Dependent ◽  
Random Parameters ◽  
Order Of Accuracy ◽  
Kummer Functions ◽  
New Methods ◽  
Large Systems

In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 105 different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.

scholarly journals New Stable, Explicit, Shifted-Hopscotch Algorithms for the Heat Equation

2021 ◽  
Vol 26 (3) ◽  
pp. 61
Author(s):  
Ádám Nagy ◽  
Mahmoud Saleh ◽  
Issa Omle ◽  
Humam Kareem ◽  
Endre Kovács
Keyword(s):  
Heat Equation ◽  
Numerical Algorithms ◽  
Nonlinear Problems ◽  
Half Time ◽  
Random Parameters ◽  
Time Step ◽  
Order Accuracy ◽  
New Methods ◽  
Large Systems ◽  
Short Time

Our goal was to find more effective numerical algorithms to solve the heat or diffusion equation. We created new five-stage algorithms by shifting the time of the odd cells in the well-known odd-even hopscotch algorithm by a half time step and applied different formulas in different stages. First, we tested 105 = 100,000 different algorithm combinations in case of small systems with random parameters, and then examined the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. These tests helped us find the top five combinations, and showed that these new methods are, indeed, effective since quite accurate and reliable results were obtained in a very short time. After this, we verified these five methods by reproducing a recently found non-conventional analytical solution of the heat equation, then we demonstrated that the methods worked for nonlinear problems by solving Fisher’s equation. We analytically proved that the methods had second-order accuracy, and also showed that one of the five methods was positivity preserving and the others also had good stability properties.

scholarly journals Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3308
Author(s):  
Humam Kareem Jalghaf ◽  
Endre Kovács ◽  
János Majár ◽  
Ádám Nagy ◽  
Ali Habeeb Askar
Keyword(s):  
Finite Difference ◽  
Initial Conditions ◽  
Finite Difference Methods ◽  
New Method ◽  
Diffusion Reaction ◽  
Fourth Power ◽  
Random Parameters ◽  
Diffusion Term ◽  
Order Of Accuracy ◽  
Large Systems

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well.

Analytical Solution of One Dimension Time Dependent Advection- Diffusion Equation

2016 ◽  
Vol 4 (2) ◽  
pp. 67-73
Author(s):  
A. A. Marrouf ◽  
Maha S. El-Otaify ◽  
Adel S. Mohamed ◽  
Galal Ismail ◽  
Khaled S. M. Essa
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Time Dependent ◽  
One Dimension ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

scholarly journals Power Spectrum and Diffusion of the Amari Neural Field

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 134
Author(s):  
Luca Salasnich
Keyword(s):  
Power Spectrum ◽  
Diffusion Equation ◽  
Analytical Solutions ◽  
Initial Conditions ◽  
Analytical Formula ◽  
Reaction Diffusion ◽  
Space Time ◽  
Neural Field ◽  
Time Dependent ◽  
Reaction Diffusion Equation

We study the power spectrum of a space-time dependent neural field which describes the average membrane potential of neurons in a single layer. This neural field is modelled by a dissipative integro-differential equation, the so-called Amari equation. By considering a small perturbation with respect to a stationary and uniform configuration of the neural field we derive a linearized equation which is solved for a generic external stimulus by using the Fourier transform into wavevector-freqency domain, finding an analytical formula for the power spectrum of the neural field. In addition, after proving that for large wavelengths the linearized Amari equation is equivalent to a diffusion equation which admits space-time dependent analytical solutions, we take into account the nonlinearity of the Amari equation. We find that for large wavelengths a weak nonlinearity in the Amari equation gives rise to a reaction-diffusion equation which can be formally derived from a neural action functional by introducing a dual neural field. For some initial conditions, we discuss analytical solutions of this reaction-diffusion equation.

scholarly journals New explicit algorithm based on the asymmetric hopscotch structure to solve the heat conduction equation

2021 ◽  
Vol 11 (5) ◽  
pp. 233-244
Author(s):  
Issa Omle
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Stage Structure ◽  
Two Dimensional ◽  
Random Parameters ◽  
Time Step ◽  
Acceptable Accuracy ◽  
New Methods ◽  
Large Systems ◽  
Short Time

In this paper we will consider a new four-stage structure inspired by the well-known odd-even hopscotch method to construct new schemes for the numerical solution of the two-dimensional heat or diffusion equation. In this structure the first and the last time step are halved stage and therefore the time steps are shifted compared to each other for odd and even cells. We insert 10 concrete formulas into this structure to obtain 104 different combinations. First we test all of these in case of small systems with random parameters, and then examine the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. We select the top 5 combinations, and demonstrate that these new methods are indeed effective if the goal is to produce results with acceptable accuracy in very short time.

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