GSoC 2nd week review | Qingyu's cabin

This week, I continued to commit and fix errors in adaptive defect control PR, and now the PR is ready for review!

When I was debugging the PR, I gradually noticed the importance of the nonlinear solving process in MIRK method, so I take a deep understanding of the special Newton iteration process in MIRKDC, here is a summary.

Newton iteration in MIRKDC

After the essential initialization, such as mesh, residual function and DE configuration, we use Newton iteration to solve the discrete system and get the discrete solution of the problem. However, the nonlinear system-solving routine in MIRKDC is quite different from what we have learned, what happened during this nonlinear equation-solving process? Here is a summary.

With the previous mesh and initialized discrete solution, we use the modified Newton method to solve the nonlinear system PHI(Y)=0 and get the solution approximation when the Newton correction is less than the tolerance newtol.

To begin the Newton iteration, we need to know that in the MIRKDC routine, we use a special Newton iteration to solve the nonlinear system. The modified Newton iteration routine, according to how well the iterates appear to be converging, can decide to use the damped Newton step or fixed Jacobian step to make iterate better converging.

Damped newton step

When we are taking a damped Newton step, we choose a damping factor, and provide a damped update to the current iterate , here, is the Newton correction: , here is the Jacobian in . By using the damped Newton step, the acceptability depends on if the damped Newton iterate produces a sufficient reduction to the natural criterion function. If the damped Newton step is successful, a fixed Jacobian step would be taken if this step is a full Newton step(to examine if ).

Fixed Jacobian step

The core of the fixed Jacobian step is also to provide an update to the current iterate, but the fixed Jacobian step provides an update to the current iterate , here, is the Newton correction: , but is evaluated in at some previous iterates. By using the fixed Jacobian step, the acceptability depends on if the new iterate also produces a sufficient reduction to the natural criterion function. If the fixed Jacobian step is successful, the routine would try another fixed Jacobian step.

We need to note that the discrete stages k_discrete for the afterward computing are computed in the process of modified Newton iterations, it is a by-product while we solve the nonlinear equations. k_discrete has a size of , where s is the stage for the given MIRK method.