This paper proposes a numerical treatment for physical systems where perturbations of solutions evolve on a much faster time scale than the solution itself, resulting in heavily, unevenly distributed eigenvalues that would cause difficulty (known as stiffness) when numerical solutions are sought.

The main idea proposed is to apply explicit difference schemes (with respect to time) with less restricted stability conditions to the linear portion of a high frequency equation, which thereby supports the mitigation of the stiffness of the linear operator, and hence achieves a balance between accuracy and stability through the choice of the so-called tuning parameter (the &bgr; below). In a bit more detail: for the viscous Burger’s equation u_t - u_xx - (1/2) (u²)_x = 0, using a change of variable w = e^{a t}u, and applying a predictor-corrector process, with, for example, a Forward-Euler scheme for temporal discretization of the linear portion of the equation, results in an unconditionally stable scheme for appropriately chosen parameter a. In conjunction, the separation of scales in finite differences, provided by the so-called incremental unknowns method, is then applied.

The stability analysis in the linear case is given in the paper. Numerical examples are presented in solving Burger’s equation, with uniform and nonuniform meshes in one and two dimensions, with Forward-Euler and second order Adams-Bashforth schemes.

The benefit of the new approach is that it has the same stability as seen in semi-implicit schemes, yet it can be solved explicitly, hence improving central processing unit (CPU) efficiency. What was proposed should be viewed more as a framework than as a particular scheme, since the idea can be easily adapted to solving, for example, an incompressible Navier-Stokes equation, reducing computational cost.