NEW METHOD FOR CHARACTERISTIC EVOLUTIONS IN NUMERICAL RELATIVITY

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
ZHOUJIAN CAO
Keyword(s):  
Cauchy Problem ◽  
Finite Difference ◽  
Differential Equations ◽  
Numerical Relativity ◽  
Einstein Equations ◽  
New Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Special Difference

The main task of numerical relativity is to solve Einstein equations with the aid of supercomputer. There are two main schemes to write Einstein equations explicitly as differential equations. One is based on 3 + 1 decomposition and reduces the Einstein equations to a Cauchy problem. The another takes the advantage of the characteristic property of Einstein equations and reduces them to a set of ordinary differential equations. The latter scheme is called characteristic formalism which is free of constraint equations in contrast to the corresponding Cauchy problem. Till now there is only one well developed code (PITT code) for characteristic formalism. In PITT code, special finite difference algorithm is adopted for the numerical calculation. And it is this special difference algorithm that restricts the numerical accuracy order to second-order. In addition, this special difference algorithm makes the popular Runge–Kutta method used in Cauchy problem not available. In this paper, we modify the equations for characteristic formalism. Based on our new set of equations, we can use usual finite difference method as done in usual Cauchy evolution. And Runge–Kutta method can also be adopted naturally. We develop a set of code in the framework of AMSS-NCKU code based on our new method and some numerical tests are done.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

scholarly journals Application of Differential Transformation Method for Solving HIV Model with Anti-Viral Treatment

2020 ◽  
Vol 14 (3) ◽  
pp. 378-388
Author(s):  
Esther Y. Bunga ◽  
Meksianis Z. Ndii
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Kutta Method ◽  
System Of Differential Equations ◽  
Time Step ◽  
Differential Transformation ◽  
Runge Kutta Method ◽  
Runge Kutta

Mathematical models have been widely used to understand complex phenomena. Generally, the model is in the form of system of differential equations. However, when the model becomes complex, analytical solutions are not easily found and hence a numerical approach has been used. A number of numerical schemes such as Euler, Runge-Kutta, and Finite Difference Scheme are generally used. There are also alternative numerical methods that can be used to solve system of differential equations such as the nonstandard finite difference scheme (NSFDS), the Adomian decomposition method (ADM), Variation iteration method (VIM), and the differential transformation method (DTM). In this paper, we apply the differential transformation method (DTM)  to solve system of differential equations. The DTM is semi-analytical numerical technique to solve the system of differential equations and provides an iterative procedure to obtain the power series of the solution in terms of initial value parameters.. In this paper, we present a mathematical model of HIV with antiviral treatment and construct a numerical scheme based on the differential transformation method (DTM) for solving the model. The results are compared to that of Runge-Kutta method. We find a good agreement of the DTM and the Runge-Kutta method for smaller time step but it fails in the large time step.

A modified Runge-Kutta method

SIMULATION ◽  
1968 ◽  
Vol 10 (5) ◽  
pp. 221-223 ◽  
Author(s):  
A.S. Chai
Keyword(s):  
Error Estimate ◽  
Linear Combination ◽  
Experimental Results ◽  
The Other ◽  
New Method ◽  
Kutta Method ◽  
Starting Values ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
A Minor

It is possible to replace k2 in a 4th-order Runge-Kutta for mula (also Nth-order 3 ≤ N ≤ 5) by a linear combination of k1 and the ki's in the last step, using the same procedure for computing the other ki's and y as in the standard R-K method. The advantages of the new method are: It re quires one less derivative evaluation, provides an error estimate at each step, gives more accurate results, and needs a minor change to switch to the RK to obtain the starting values. Experimental results are shown in verification of the for mula.

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