Jha Navnit, Singh Bhagat
Faculty of Mathematics and Computer Science, South Asian University, Maidan Garhi, New Delhi 110 068, India.
Department of Mathematics, G.D. Goenka University, Sohna 122103, India.
MethodsX. 2023 Oct 9;11:102424. doi: 10.1016/j.mex.2023.102424. eCollection 2023 Dec.
The analysis of nonlinear elliptic PDEs representing stationary convection-dominated diffusion equation, Sine-Gordon equation, Helmholtz equation, and heat exchange diffusion model in a battery often lacks in closed-form solutions. For the long-term behaviour and to assess the quantitative behaviour of the model, numerical treatment is necessary. A novel numerical approach based on arithmetic average compact discretization employing a quasi-variable grid network is proposed for a wide class of nonlinear three-dimensional elliptic PDEs. The method's key benefit is that it applies to singular models and only needs nineteen-point grids with seven functional approximations. Additionally, the suggested method disseminates the truncation error across the domain, which is unrealistic for finite-difference discretization with a fixed step length of grid points. Often, small diffusion anticipates strong oscillation, and tuning the grid stretching parameter helps error dispersion over the domain. The scheme is examined for maximal error bounds and convergence property with the help of a monotone matrix and its irreducible character. The metrics of solution accuracies, mainly root-mean-squared and absolute errors alongside numerical convergence rate, are inspected by different types of variable coefficients, singular and non-singular 3D elliptic PDEs appearing in a convection-diffusion phenomenon. The performance of the numerical solution corroborates the fourth-order convergence on a quasi-variable grid network.
对表示稳态对流主导扩散方程、正弦 - 戈登方程、亥姆霍兹方程以及电池中的热交换扩散模型的非线性椭圆型偏微分方程的分析通常缺乏封闭形式的解。为了研究模型的长期行为并评估其定量特性，数值处理是必要的。针对一类广泛的非线性三维椭圆型偏微分方程，提出了一种基于算术平均紧致离散化并采用准可变网格网络的新颖数值方法。该方法的关键优势在于它适用于奇异模型，并且只需要具有七个函数近似的十九点网格。此外，所提出的方法将截断误差分散到整个区域，这对于具有固定网格点步长的有限差分离散化来说是不现实的。通常，小扩散预示着强烈振荡，调整网格拉伸参数有助于在整个区域上分散误差。借助单调矩阵及其不可约特性，对该格式进行了最大误差界和收敛性的检验。通过对流扩散现象中出现的不同类型的可变系数、奇异和非奇异三维椭圆型偏微分方程，考察了解的精度指标，主要是均方根误差和绝对误差以及数值收敛率。数值解的性能证实了在准可变网格网络上的四阶收敛性。
