Partial differential equations are ubiquitous in this day and age, finding many uses in STEM fields. An important class of equations, arising often in practice, is the hyperbolic semilinear one. In this paper, a family of implicit numerical schemes for solving such partial differential equations is derived, utilising finite differences and tridiagonal sweep. Results for several semilinear partial differential equations are presented and analysed. Equations with power-law, trigonometric and hyperbolic nonlinearities are considered, some famous examples being the sine-Gordon and sinh-Gordon equations. The numerical findings are compared and an optimal subset of the family is found.