In this paper, a new implicit-explicit local method with an arbitrary order is produced for stiff initial value problems. Here, a general method for one-step time integrations has been created, considering a direction free approach for integrations leading to a numerical method with parameter-based stability preservation. Adaptive procedures depending on the problem types for the current method are explained with the help of local error estimates to minimize the computational cost. Priority error analysis of the current method is made, and order conditions are presented in terms of direction parameters. Stability analysis of the method is performed for both scalar equations and systems of differential equations. The currently produced parameter-based method has been proven to provide A-stability, for 0.5<\theta<1, in various orders. The present method has been shown to be a very good option for addressing a wide range of initial value problems through numerical experiments. It can be seen as a significant contribution that the Susceptible-Exposed-Infected-Recovered equation system parameterized for the COVID-19 pandemic has been integrated with the present method and stability properties of the method have been tested on this stiff model and significant results are produced. Some challenging stiff behaviours represented by the nonlinear Duffing equation, Robertson chemical system, and van der Pol equation have also been integrated, and the results revealed that the current algorithm produces much more reliable results than numerical techniques in the literature.
翻译:在本文中,针对初始值的严酷问题,产生了一种新的隐含、含任意顺序的局部方法。在这里,为一步时间整合制定了一种一般方法,考虑采用一种无方向的整合方法,以得出基于参数的稳定性保存的数值方法。根据当前方法的问题类型解释适应程序,以当地误差估计为根据,以尽量减少计算成本。对当前方法进行了优先误差分析,按方向参数提出了顺序条件。对标价方程和差异方程系统进行了稳定分析。目前制作的基于参数的集成方法已被证明能够提供A的可耐用性,在各种顺序中为0.5 ⁇ theta < 1提供A的可耐性。目前的方法已被证明是一个非常好的选择,通过数字实验解决一系列初步价值问题。可以认为,对当前方法进行优先误差分析,从方向参数参数的角度对当前方法进行排序的公式系统进行了稳定分析。对当前方法的方法和差异方程的稳定性特性进行了分析。目前采用的基于参数的方法已被证明为A的可耐用性,在这种硬性模型上提供了A-thetata < 1,并且通过不具有相当挑战性的方程的方程的计算结果。