4 th order Embedded Boundary FDTD algorithm for Maxwell Equations Lingling Wu, Stony Brook University Roman Samulyak, BNL Tianshi Lu, BNL Application collaborators: J. Cary, TechX

Research Goals Develop 4 th order accurate in both space and time embedded boundary FDTD algorithms Develop stable and accurate algorithm for coupling with particles Evaluate accuracy and performance of different versions of algorithms Assist in the implementation of the most promising algorithms in TechX EM software Motivation: Provide robust, accurate and scalable algorithms for the simulation of EM waves coupled to charge particles in geometrically complex RF cavities).

Advantages of High order Embedded Boundary Approach Simple approach to geometrically complex domains Avoids the generation of complex conforming FE meshes Improves the stability of unstructured schemes Preserves conservation properties Accuracy in coupling to particles Rapid convergence Scalability to 1000s processors

Main Ideas: Spatial Discretization Preserves the main simplification of the embedded boundary method: magnetic field is located in centers of the regular grid cells (even if cells are cut by the boundary) Fourth order spatial discretization of derivatives (3rd order one-sided derivatives near boundaries) Calculation of cell volume using curvature information Merge of small volume cut cells

4 th order symplectic time integrator using Yoshidas method based on three Leapfrog steps. In general Main Ideas: Temporal Discretization Preserves the symplectic property of the Leapfrog scheme For comparison, the Runge-Kutta integrator is not symplectic Smaller storage compared to RK4, Yoshida / RK4 =1 / 4 Minor reduction of CFL condition compared to RK4 Fixed T, operation time ratio is Yoshida / RK4=1.26 / 1

Current progress Performed rigorous theoretical analysis of various components of the algorithm Developed FDTD code prototype Fully implemented and tested: 2 nd order embedded boundary algorithm based on Leapfrog time stepping 4 th order Yoshida time integrator 4 th order FDTD scheme for regular geometries Implementation 4 th order embedded boundary algorithm is in progress

Future work Complete implementation of 4 th order embedded boundary algorithm Evaluate other versions of the algorithm (unconditionally stable) Develop and implement algorithm for coupling with particles Assist in the implementation of algorithms in TechX EM software

Evaluation of other versions of the algorithm Operator split techniques based on an implicit Crank–Nicolson discretization of elementary systems and Yoshidas time loop. Split the equations using the ADI idea and solve the split equations by Crank-Nicolson scheme and improve the order by Yoshida technique. ADI is first order for time and can be improved to 2 nd order by strang split and then improved to 4 th order by Yoshida technique. Crank-Nicolson is 2 nd order for space and unconditional stable. It also shows 2 nd order for time for the split equations. We can improve the spatial order by using longer stencil and keep the 2 nd order for time. This will allow us to avoid merging small cut cells in order to satisfy the stability condition. Merging small cells introduces error.

Thank you !