摘要：This talk consists of two interactive components. First, we will pay an attention to
classical splitting methods, such as the non-exponential ADI and exponential LOD
methods, and explore their modernizations. Then we will focus at interesting issues
involving the design and analysis of highly-effective and highly-efficient finite difference
methods for solving singular Kawarada equations which are fundamental in numerical
combustion applications. We will outline the physical background of the quenching
phenomena involved. Adaptive splitting approaches will be introduced. Numerical
analysis on their monotonicity, convergence and stability will be discussed. We will also
present ideas of the latest exponential evolving grid development inspired by moving
grid strategies which can be extended for solving multiphysics equations with similar
singularities from studies of biophysics, oil pipeline decay detections, cancer treatments,
and laser-materials interactions. Certain stochastic inferences will be mentioned.
Potentials of further investigations and collaborations will be discussed.