A New Slope Averaging Technique in Heun's Method: Combining Harmonic and Root Mean Square Means
Abstract
This paper introduces a modified version of Heun's method for solving initial value problems (IVPs) in ordinary differential equations (ODEs). The proposed method replaces the standard arithmetic mean used for the average slope in Heun's method with a novel combination of harmonic mean and root mean square (RMS) for the slopes of the tangent lines. This alteration is designed to enhance the accuracy and performance of the method. Through theoretical analysis, we establish that the modified method retains stability and consistency properties, crucial for reliable numerical solutions. Extensive numerical experiments demonstrate that the method performs well across a range of step sizes, showing notable improvements in accuracy for both small and large step sizes when compared to the standard Heun's method. These results suggest that the new approach offers a robust alternative for solving IVPs, particularly in scenarios where adaptive step sizing or high precision is required.
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