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.

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 A New Trigonometrically Fitted Two-Derivative Runge-Kutta Method for the Numerical Solution of the Schrödinger Equation and Related Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yanwei Zhang ◽  
Haitao Che ◽  
Yonglei Fang ◽  
Xiong You
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
Recent Literature ◽  
New Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Radial Schrödinger Equation ◽  
Runge Kutta ◽  
Fifth Order

A new trigonometrically fitted fifth-order two-derivative Runge-Kutta method with variable nodes is developed for the numerical solution of the radial Schrödinger equation and related oscillatory problems. Linear stability and phase properties of the new method are examined. Numerical results are reported to show the robustness and competence of the new method compared with some highly efficient methods in the recent literature.

Modeling the Effects of Chemotherapy and Immunotherapy on Tumor Growth

2021 ◽  
Vol 17 (12) ◽  
pp. 2505-2518
Author(s):  
Sara El Haout ◽  
Maymunah Fatani ◽  
Nadia Abu Farha ◽  
Nour AlSawaftah ◽  
Maruf Mortula ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Cd8 T Cells ◽  
The Other ◽  
Kutta Method ◽  
Cancer Treatments ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Combined Intervention ◽  
Combined Interventions

Mathematical modeling has been used to simulate the interaction of chemotherapy and immunotherapy drugs intervention with the dynamics of tumor cells growth. This work studies the interaction of cells in the immune system, such as the natural killer, dendritic, and cytotoxic CD8+ T cells, with chemotherapy. Four different cases were considered in the simulation: no drug intervention, independent interventions (either chemotherapy or immunotherapy), and combined interventions of chemotherapy and immunotherapy. The system of ordinary differential equations was initially solved using the Runge-Kutta method and compared with two additional methods: the Explicit Euler and Heun’s methods. Results showed that the combined intervention is more effective compared to the other cases. In addition, when compared with Runge-Kutta, the Heun’s method presented a better accuracy than the Explicit Euler technique. The proposed mathematical model can be used as a tool to improve cancer treatments and targeted therapy.

scholarly journals A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
M. Mechee ◽  
N. Senu ◽  
F. Ismail ◽  
B. Nikouravan ◽  
Z. Siri
Keyword(s):  
Thin Film ◽  
Differential Equation ◽  
Film Flow ◽  
New Method ◽  
Kutta Method ◽  
Thin Film Flow ◽  
Third Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fifth Order

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

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 An Optimized Runge-Kutta Method for the Numerical Solution of the Radial Schrödinger Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Qinghe Ming ◽  
Yanping Yang ◽  
Yonglei Fang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Efficiency ◽  
New Method ◽  
Kutta Method ◽  
Saxon Potential ◽  
First Derivative ◽  
Runge Kutta Method ◽  
Radial Schrödinger Equation ◽  
Runge Kutta

An optimized explicit modified Runge-Kutta (RK) method for the numerical integration of the radial Schrödinger equation is presented in this paper. This method has frequency-depending coefficients with vanishing dispersion, dissipation, and the first derivative of dispersion. Stability and phase analysis of the new method are examined. The numerical results in the integration of the radial Schrödinger equation with the Woods-Saxon potential are reported to show the high efficiency of the new method.

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