Fuzzy transform algorithm based on high-resolution compact discretization for three-dimensional nonlinear elliptic PDEs and convection-diffusion equations

Abstract This paper deals with a high-resolution algorithm that engages fuzzy transform to solve three-dimensional nonlinear elliptic partial differential equations. The scheme approximates the fuzzy components, which estimate fourth-order accurate solutions at the interior mesh points of the solution domain. The fuzzy components and triangular base functions will be approximated with a nineteen-point linear combination of solution values and related to exact solutions by a linear system. Such an arrangement along with compact discretization yields a block tri-diagonal Jacobian matrix, and an iterative solver can efficiently compute them. The convergence analysis and error bound of the scheme are examined in detail. The method provides an order-preserving solution and applies to a comprehensive class of partial differential equations with nonlinear first-order partial derivatives. Numerical simulations with Helmholtz equation, advection-diffusion-reaction equation, and nonlinear elliptic Sine-Gordan equation corroborate the utility, convergence rate and enhance solution accuracy by employing a new scheme.