Splitting schemes for nonlinear parabolic problems

Abstract: This thesis is based on five papers, which all analyse different aspects of splitting schemes when applied to nonlinear parabolic problems. These numerical methods are frequently used when a problem has a natural decomposition into two or more parts, as the computational cost may then be significantly decreased compared to other methods. There are two prominent themes in the thesis; the first concerns convergence order analysis, while the second focuses on structure preservation. To motivate the first theme, we note that even if a method has been shown to converge it might be that the speed of convergence is arbitrarily slow. As such a method is unusable in practice we see that it is essential to prove convergence orders. However, those studies that present such error analyses in the fully nonlinear setting typically assume more regularity of the solution than what should be expected. In this context, we present a convergence order analysis for a class of splitting schemes which, importantly, does not require any artificial regularity assumptions. This analysis is carried out in the setting of m-dissipative operators, which includes a large number of interesting problem classes. As demonstrated by the first three papers, the theory can be applied to such diverse problems as nonlinear reaction-diffusion systems, nonlinear parabolic problems with delay, as well as differential Riccati equations. Within the second theme of structure preservation, an in-depth study of operator-valued differential Riccati equations has been carried out. In such equations it is desirable for a numerical method to produce positive semi-definite approximations. Further, it is essential that an implementation can utilize the problem-inherent property of low rank. As shown in the last three papers, both these features are readily satisfied for various splitting schemes. Since these are additionally less costly than existing comparable methods, they constitute a particularly competitive choice for such problems.