Unconditionally stable meshless method for solving 3D transient electromagnetic problems based on LOD scheme

Abstract

One of the main problems of the conventional meshless methods in the time domain is their conditional stability. Up to now, some algorithms have been introduced to remove the stability condition, like alternating-direction-implicit (ADI) and locally one-dimensional (LOD) schemes. ADI meshless method has been investigated completely, but there is no report to exploit the full efficiency of the LOD meshless method. So, in this study, LOD scheme is employed to a meshless method for solving three-dimensional transient electromagnetic problems. The initial form of the proposed meshless method solves time-domain Maxwell's equations in three sub-steps, so it is called LOD3. LOD3 meshless method has first-order temporal accuracy. Moreover, LOD5 meshless method is introduced to upgrade temporal accuracy. Also, the results show that LOD5 meshless method is more accurate than ADI and LOD3 meshless methods. The accuracy of the proposed method is due to two main factors; Crank–Nicolson scheme and solving the equation in five sub-steps instead of one sub-step. Stability and accuracy of the proposed method are assessed through numerical experiment. Also, the unconditional stability of the proposed method is proved, analytically.