Moving mesh methods for blowup in reaction–diffusion equations with traveling heat source

A numerical investigation of blow-up in reaction–diffusion problems with traveling heat sources

“…But some modification should be carefully made when discretizing the terms u x and u xx at the irregular mesh point. By constructing a smooth function, five different cases of the discretization are developed in [15]. Different from them, we add auxiliary mesh Fig.…”

“…Different from them, we add auxiliary mesh Fig. 1) discussed in [15] will not occur. Thus, only the following three cases of the discretization are considered in the new moving mesh method.…”

“…With respect to the fixed mesh method, moving mesh method achieves the significant improvements in the accuracy and efficiency [7][8][9]. Recently, MMPDEs (moving mesh PDEs) approach is developed in [10] and has been successfully applied in a few blow-up partial differential equations [11][12][13][14][15][16]. In [10], several satisfactory mesh equations called MMPDEs are presented.…”

“…There is no doubt about the auxiliary points may help to accurately capture the singularity. Moreover, we simplify the investigation cases of the discretization in [15] and are adequate to solve the problem (1) with two moving sources. However, unlike MMPDEs in [15], the new method can be viewed as a combination of moving mesh method and a special adaptive method.…”

“…Moreover, we simplify the investigation cases of the discretization in [15] and are adequate to solve the problem (1) with two moving sources. However, unlike MMPDEs in [15], the new method can be viewed as a combination of moving mesh method and a special adaptive method. The critical auxiliary points, which are introduced in Section 2, only exist at the current temporal level and do not move into the next temporal level.…”