TOPIC : Efficient and Accurate Local Time Stepping

Algorithms for Multi-rate Problems

By Dr. Jianping Zhu

Mississippi State University

Summary and Comments : By Lee Emmanwori

A) Dr. Zhu introduced the Efficient and Accurate Local Time Stepping Algorithms

for Multi-rate Problems with heat equation :

U(0,t) = f(t)

U(x,0) = g(t)

U(x,0) = h(x)

He explained Explicit and Implicit integration of the heat equation using the local time stepping algorithms.

B)Dr. Zhu compared explicit versus implicit methods of this algorithm. He pointed out that 1) Explicit Method are:

easy for implementation

parallelization

nonlinear problems

and only conditionally stable r <=1/2

2) Implicit Method are:

Unconditionally stable

Difficult for implementation

parallelization

nonlinear problems

3) Both methods have been widely used

4) Traditional time stepping methods use a uniform delta-t

C) Dr. Zhu explained that the alternative is to use accurate local time stepping method whose basic idea is that the solution at each grid point is advanced using the

time step size determined by the local stability criteria of r<=1/2.

D) Dr. Zhu explained applying local time stepping algorithms to implicit method using -- Standard Crank-Nicholson

nonlinear algebraic equations

Newton's method

Block matrix

E) Finally, Dr. Zhu compared coupled systems versus decoupled systems. He also presented numerical results to demonstrate significant improvement in computational efficiency as a result of using efficient and accurate local time stepping algorithms for multi-rate problems.