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.

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 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.

ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS

2003 ◽  
Vol 13 (05) ◽  
pp. 613-644 ◽  
Author(s):  
ESPEN ROBSTAD JAKOBSEN
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Optimal Stopping ◽  
Initial Conditions ◽  
Finite Difference Methods ◽  
Approximation Schemes ◽  
Dynamic Programming Principle ◽  
Bellman Equations ◽  
Difficult Time ◽  
Difference Methods

We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

scholarly journals Fast and simple method for pricing exotic options using Gauss–Hermite quadrature on a cubic spline interpolation

2014 ◽  
Vol 01 (04) ◽  
pp. 1450033 ◽  
Author(s):  
Xiaolin Luo ◽  
Pavel V. Shevchenko
Keyword(s):  
Finite Difference ◽  
Spline Interpolation ◽  
Finite Difference Methods ◽  
Cubic Spline ◽  
New Method ◽  
Exotic Options ◽  
Simple Method ◽  
Cubic Spline Interpolation ◽  
Difference Methods ◽  
Path Dependent

There is a vast literature on numerical valuation of exotic options using Monte Carlo (MC), binomial and trinomial trees, and finite difference methods. When transition density of the underlying asset or its moments are known in closed form, it can be convenient and more efficient to utilize direct integration methods to calculate the required option price expectations in a backward time-stepping algorithm. This paper presents a simple, robust and efficient algorithm that can be applied for pricing many exotic options by computing the expectations using Gauss–Hermite integration quadrature applied on a cubic spline interpolation. The algorithm is fully explicit but does not suffer the inherent instability of the explicit finite difference counterpart. A "free" bonus of the algorithm is that it already contains the function for fast and accurate interpolation of multiple solutions required by many discretely monitored path dependent options. For illustrations, we present examples of pricing a series of American options with either Bermudan or continuous exercise features, and a series of exotic path-dependent options of target accumulation redemption note (TARN). Results of the new method are compared with MC and finite difference methods, including some of the most advanced or best known finite difference algorithms in the literature. The comparison shows that, despite its simplicity, the new method can rival with some of the best finite difference algorithms in accuracy and at the same time it is significantly faster. Virtually the same algorithm can be applied to price other path-dependent financial contracts such as Asian options and variable annuities.

scholarly journals An Invariant-Preserving Scheme for the Viscous Burgers- Poisson System

Computation ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 115
Author(s):  
Chayapa Darayon ◽  
Morrakot Khebchareon ◽  
Nattapol Ploymaklam
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Second Order ◽  
Water Model ◽  
Poisson System ◽  
Shallow Water Model ◽  
Order Of Accuracy ◽  
Pointwise Error ◽  
First Time ◽  
Theoretical Results

We formulate and analyze a new finite difference scheme for a shallow water model in the form of viscous Burgers-Poisson system with periodic boundary conditions. The proposed scheme belongs to a family of three-level linearized finite difference methods. It is proved to preserve both momentum and energy in the discrete sense. In addition, we proved that the method converges uniformly and has second order of accuracy in space. The analysis given in this work is the first time a pointwise error estimation is done on a second-order finite difference operator applied to the Burgers-Poisson system. We validate our findings by performing various numerical simulations on both viscous and inviscous problems. These numerical examples show the efficacy of the proposed method and confirm the proven theoretical results.

scholarly journals Application of the Finite Differences Methods to Computation of Definite Integrals

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Multistep Methods ◽  
Finite Difference Methods ◽  
Engineering Problem ◽  
Stable Method ◽  
Order Of Accuracy ◽  
Definite Integrals ◽  
The Finite Difference Method

- It is known that one of the popular methods constructed to solve the scientific and engineering problem is the finite difference method studied beginning from the known and famous scientists as the Newton, Leibniz, Euler and etc. One of the first applications of the finite difference method is defined as the conception of the derivatives for the continuous function. These methods have been successfully used in solving of the differential equations at the present time. But here have investigated the application of the finite-difference methods to compute the definite integrals. Constructed the stable methods with the high order of accuracy which are applied to computation of the definite integrals. Also has been found some connection between the constructed multistep methods, the Gauss and Chebyshev methods. The received here methods have been applied to the computation of the double definite integral. For this aim, here have been investigated the double integrals for the degenerate functions. The advantages of these methods are illustrated by using the known model integral, which has been solved by using the stable method with the degree p=6.

scholarly journals A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain

2014 ◽  
Vol 16 (2) ◽  
pp. 345-358 ◽  
Author(s):  
Jan Nordström ◽  
Qaisar Abbas ◽  
Brittany A. Erickson ◽  
Hannes Frenander
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Finite Difference Methods ◽  
Difference Operators ◽  
Hyperbolic Problems ◽  
Order Of Accuracy ◽  
Weak Boundary ◽  
Difference Methods ◽  
Weak Boundary Conditions ◽  
High Order Finite Difference

AbstractA new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied near boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries. The new boundary procedure is cheap, easy to implement and suitable for all numerical methods, not only finite difference methods, that employ weak boundary conditions. Numerical results that corroborate the analysis are presented.

Taylor vortices between two concentric rotating spheres

1982 ◽  
Vol 119 ◽  
pp. 1-25 ◽  
Author(s):  
Fritz Bartels
Keyword(s):  
Reynolds Number ◽  
Finite Difference ◽  
Flow Field ◽  
Critical Reynolds Number ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Finite Difference Methods ◽  
Taylor Vortices ◽  
Concentric Spheres ◽  
Inner Sphere

The laminar viscous flow in the gap between two concentric spheres is investigated for a rotating inner sphere. The solution is obtained by solving the Navier-Stokes equations by means of finite-difference techniques, where the equations are restricted to axially symmetric flows. The flow field is hydrodynamically unstable above a critical Reynolds number. This investigation indicates that the critical Reynolds number beyond which Taylor vortices appear is slightly higher in a spherical gap than for the flow between concentric cylinders. The formation of Taylor vortices could be observed only for small gap widths s ≤ 0·17. The final state of the flow field depends on the initial conditions and the acceleration of the inner sphere. Steady and unsteady flow modes are predicted for various Reynolds numbers and gap widths. The results are in agreement with experiment if certain accuracy conditions of the finite-difference methods are satisfied. It is seen that the equatorial symmetry is of great importance for the development of the Taylor vortices in the gap.

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.

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