In this work, we propose a new class of $A$-stable diagonally implicit four stage Runge–Kutta (R–K) methods with minimal dissipation and optimally low dispersion error. These schemes obtained by minimizing both amplification and phase error enjoy fourth order of accuracy and are suitable for stiff systems. Emphasis here is to outline an algorithm that can be used to develop diagonally implicit R–K methods of diverse stages having low-dissipation low-dispersion virtues while retaining, to a large extent, inherent stability and high accuracy. This algorithm is subsequently applied to propose two, three and four stage diagonally implicit R–K schemes. One and two dimensional, linear and non-linear propagation problems are numerically tackled at a relatively higher CFL number and a comprehensive comparison is carried out with other stable diagonally implicit schemes available in the literature to exhibit benefits of optimization.

在这项工作中，我们提出了一个新的 $\mathrm{一种}$稳定的对角隐式四阶段Runge-Kutta（R-K）方法，具有最小的耗散和最佳的低色散误差。通过最小化放大和相位误差获得的这些方案具有四级精度，适用于刚性系统。这里的重点是概述一种算法，该算法可用于开发具有低耗散，低散度优点的不同阶段的对角隐式R–K方法，同时在很大程度上保留固有的稳定性和高精度。此算法随后应用于提出两，三和四级对角线隐式R–K方案。一维和二维