SC740 SEMINAR REPORT 10 for Frederick L. Jones

PRESENTER: Dr. Qin Sheng

OVERVIEW

Dr. Qin Sheng’s presentation focused on numerical solution techniques for one very important variation of the Heat Equation, i.e., Degenerate Semilinear Combustion Quenching Equations. The numerical solution techniques that Dr. Sheng presented are certain specially formulated Adaptive Grid methods.

The Adaptive Grid methods that Dr. Sheng described were modified arc-length procedures for robust adjustments of the discretizations in the spatial and/or the temporal directions. The numerical solution generated converges monotonically up to the combustion-quenching point.

DEGENERATE SEMILINEAR COMBUSTION QUENCHING EQUATIONS

Dr. Qin Sheng began his presentation with a description one very important variation of the Heat Equation, i.e., Degenerate Semilinear Combustion Quenching Equations. Dr Sheng’s description of degenerate semilinear combustion-quenching equations focused on the following differential equation (d.e.):

u( 0,t ) = u( 1,t ) = 0

0 < s < 1, 0 < p(x) < b

u |_t=0 = 0 and u |_T = 0

The focus of Dr. Sheng’s description of degenerate semilinear combustion-quenching equations was on the parameter s in the d.e. The parameter s determines the switch between states in the d.e., i.e. s is the combustion quenching point. Dr. Sheng stated that the value of s used was 1 / ( 2 Ö 2 ) or » 0.6535.

ADAPTIVE GRID COMPUTATIONS

When Dr. Qin Sheng began discussing Adaptive Grid Computations, he noted that Adaptive Grid or Moving Mesh methods are becoming valuable computational techniques in numerical solutions of d.e.’s such as the semilinear combustion-quenching d.e.’s. Dr. Sheng noted that there is a need for having a variable step size in numerical solutions; however, he also noted that we need constraints on step sizes for stability.

The Adaptive Grid methods that Dr. Sheng described were modified arc-length procedures for robust adjustments of the discretizations in the spatial and/or the temporal directions. The arc-length adaptive mechanism in time is based on the rate function v_t rather than u or u_t. The main result of Dr. Sheng’s use of the Adaptive Grid Method was his acceptance of a smaller grid size than he had initially planned to use.

The Adaptive Grid method was used with the two-stage Runge-Kutta method. The numerical solution generated converges monotonically up to the combustion-quenching point. Dr. Sheng noted that there is a choice of proper monitor functions; however, he noted that the function u_t blows up rather than the function u.

SUMMARY AND CONCLUSIONS

Dr. Qin Sheng’s presentation focused on numerical solution techniques for one very important variation of the Heat Equation, i.e., Degenerate Semilinear Combustion Quenching Equations. The numerical solution techniques that Dr. Sheng presented are certain specially formulated Adaptive Grid methods.

The Adaptive Grid methods that Dr. Sheng described were modified arc-length procedures for robust adjustments of the discretizations in the spatial and/or the temporal directions. The numerical solution generated converges monotonically up to the combustion-quenching point. The main result of Dr. Sheng’s use of the Adaptive Grid Method was his acceptance of a smaller grid size than he had initially planned to use.

The only question I have concerning the use of a smaller grid size than was initially planned is that there are tradeoffs between the accuracy gained by a smaller grid size and the almost inevitable increase in round-off errors. This fact, however, does not reduce the importance of the Adaptive Grid method that Dr. Sheng described.