Spatial attrition modeling: Stability conditions for a 2D+t2D+t FD formulation

A new general formulation for the spatial modeling of combat is presented, where the main drivers are movement attitudes and struggle evolution. This model in its simplest form is represented by a linear set of two coupled partial differential equations for two independent functions of the space and time variables. Even though the problem has a linear shape, non-negative values for the two functions render this problem as nonlinear. In contrast with other attempts, this model ensures stability and theoretical consistency with the original Lanchester Equations, allowing for a better understanding and interpretation of the spatial modeling. As a numerical illustration a simple combat situation is developed. The model is calibrated to simulate different troop movement tactics that allow an invader force to provoke maximum damage at a minimum cost. The analysis provided here reviews the trade-off between spatial grid and time stepping for attrition cases and then extends it to a new method for guaranteeing good numerical behavior when the solution is expected to grow along the time variable. There is a wide variety of spatial problems that could benefit from this analysis.