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Research Article
Fastest - A New High Order FV Method Dynamically Locally Self h-Adaptive for Convective-diffusive Problems
Issue:
Volume 14, Issue 4, August 2025
Pages:
69-92
Received:
8 June 2025
Accepted:
30 June 2025
Published:
15 August 2025
Abstract: Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.
Abstract: Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory a...
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Research Article
A Meshless Method for Solving 1D Transport Equation on a Curve of ℝ2
Aguemon Wiwegnon Uriel-Longin*,
Kalivogui Siba,
Coulibaly Bakary D
Issue:
Volume 14, Issue 4, August 2025
Pages:
93-101
Received:
16 June 2025
Accepted:
7 July 2025
Published:
20 August 2025
DOI:
10.11648/j.pamj.20251404.12
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Abstract: In this paper, we present the one-dimensional transport equation solving by the isogeometric method, a meshless method using Galerkin Method. This equation has been solved on a semicircle, a curve on ℝ2. The basis of approximation used for this work, is the B-splines basis. We define univariate B-splines. We look at their properties as well as b-splines curves. We calculate the numerical solution of the transport equation using the principle of Galerkin’s method. The Galerkin method approximates the solution of a differential equation by projecting the original problem onto a finite-dimensional subspace. This approach ensures that the residual defined as the error between the exact and approximate solutions, is orthogonal to this subspace. The numerical solution is calculated, using the parametrization of the domain and using the numerical integration of Gauss. Numerical tests have been presented to show the efficiency of this method.
Abstract: In this paper, we present the one-dimensional transport equation solving by the isogeometric method, a meshless method using Galerkin Method. This equation has been solved on a semicircle, a curve on ℝ2. The basis of approximation used for this work, is the B-splines basis. We define univariate B-splines. We look at their properties as well a...
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Research Article
An Interesting Property of Certain Binomial Coefficients That Leads to a Fascinating Refinement of Wilson’s Theorem
Joseph Granville Gaskin*
Issue:
Volume 14, Issue 4, August 2025
Pages:
102-105
Received:
16 June 2025
Accepted:
9 July 2025
Published:
22 August 2025
Abstract: Wilson’s Theorem states that if p is a prime number, then the product of the first (p - 1) positive integers, increased by one is divisible by p. This classical result in number theory was attributed to John Wilson by Edward Waring in 1770. The first known proof of Wilson’s Theorem was published in 1771 by the French mathematician Joseph-Louis Lagrange. Wilson’s Theorem has applications in primality testing, cryptography, and various other areas of mathematics. In this article, we shall prove that if p is a prime number, then for each natural number, j, that lies between 1 and (p - 2), the binomial coefficient (p - 1) choose j, decreased by the jth power of negative one, is divisible by p. As such, these new results can be viewed as extensions of Wilson’s Theorem. We shall also prove that if p is a prime number, then the square of the factorial of the ratio of (p - 1) and two, is congruent to either 1 modulo p or - 1 modulo p. Additionally, we shall prove that for all positive integers m bigger than or equal to 5, m is prime if and only if m divides the sum of either one or negative one and the square of the factorial of the ratio of (m - 1) and two. Next, we shall use some of the schemes developed earlier to investigate the behavior of the divisors of the sum of prime powers of relatively prime positive integers a, b. Lastly, we shall show that if p is a prime number and k is a positive integer, then the ratio of the sum of prime powers of a, b and the sum of a, b is not divisible by the kth power of p.
Abstract: Wilson’s Theorem states that if p is a prime number, then the product of the first (p - 1) positive integers, increased by one is divisible by p. This classical result in number theory was attributed to John Wilson by Edward Waring in 1770. The first known proof of Wilson’s Theorem was published in 1771 by the French mathematician Joseph-Louis Lagr...
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