Many dynamical systems in science and engineering are governed by differential equations whose solutions must remain positive and conservative. Standard numerical methods often fail to preserve these essential properties, leading to unphysical results such as negative concentrations or loss of mass. This book presents a rigorous analytical study of modified Patankar-type schemes, a class of nonlinear time-integration methods that guarantee unconditional positivity and exact conservation while achieving at least second-order accuracy. Beyond this specific class, many of the theoretical results are shown to be applicable to general nonlinear numerical methods, providing broader insight into the design of structure-preserving schemes.
