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 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.

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.

Time and space meshes adaptivity for the resolution of a semi-linear heat equation

2021 ◽  
pp. 1-10
Author(s):  
Nejmeddine Chorfi
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Order Of Convergence ◽  
Linear Parabolic Equation ◽  
Spectral Elements ◽  
Euler Scheme ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Linear Heat ◽  
Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

scholarly journals Durée de la récupération et puissance maximale anaérobie au cours de la journée

2003 ◽  
Vol 28 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Guy Falgairette ◽  
François Billaut ◽  
Sophiane Ramdani
Keyword(s):  
Peak Power ◽  
Biological Rhythms ◽  
Cycle Ergometer ◽  
Time Of Day ◽  
Total Work ◽  
Half Time ◽  
Instantaneous Power ◽  
Recovery Processes ◽  
Logarithmic Relationship ◽  
Short Time

Effects of recovery duration (2-3 s, 15 s, 30 s, 1 min, and 2 min) and time of day (9 a.m., 2 p.m., and 6 p.m.) on sprint performance were studied in 9 subjects using a cycle ergometer. The peak power (Ppeak) and the total work performed (W) were determined from changes in instantaneous power, taking into account the inertia of the flywheel. A decrease in Ppeak and W was observed after 15 s and 2-3 s recovery (p < 0.001). A logarithmic relationship (y = 3.92 ln x + 81.5; r = 0.82; n = 9) was found between Ppeak (%Ppeak of the first sprint) and the duration of the recovery (half-time = 14.3 s; SD = 7.6). Data indicated that there was no significant effect of time of day on Ppeak and W, regardless of the duration of recovery. The recovery processes occurred in a very short time and did not seem to be affected by biological rhythms. Key words: performance, diurnal variation, fatigue, ergometry, inertia

Book Reviews

2011 ◽  
Vol 49 (1) ◽  
pp. 150-150
Keyword(s):  
Risk Management ◽  
Stanford University ◽  
Conditional Correlation ◽  
New Paradigm ◽  
Dynamic Conditional Correlation ◽  
New Methods ◽  
Northwestern University ◽  
Large Systems ◽  
Correlation Performance

Jules H. van Binsbergen of Northwestern University, Stanford University, and NBER reviews “Anticipating Correlations: A New Paradigm for Risk Management” by Robert Engle. The EconLit Abstract of the reviewed work begins, “Presents a collection of new methods for estimating and forecasting correlations for large systems of assets. Discusses correlation economics; correlations in theory; models for correlation; dynamic conditional correlation; dynamic conditional correlation performance; the MacGyver method; generalize….”

scholarly journals An integral equation–based numerical method for the forced heat equation on complex domains

2020 ◽  
Vol 46 (5) ◽  
Author(s):  
Fredrik Fryklund ◽  
Mary Catherine A. Kropinski ◽  
Anna-Karin Tornberg
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Helmholtz Equation ◽  
Singular Integrals ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Elliptic Pdes ◽  
Time Step ◽  
Modified Helmholtz Equation ◽  
High Regularity

Abstract Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

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