Symbolic Explicit Solutions for 1‐Dimensional Linear Diffusive Wave Equation With Lateral Inflow and Their Applications

The diffusive wave equation, a simplified form of the Saint‐Venant equations, is extensively used in flood routing. To solve the equation, numerous methods have been developed over the years. Most of them are numerical and hence their application generally requires case‐specific modeling and analysis to ensure stable solution. For many practical routing applications, however, simpler yet accurate methods are highly desirable that do not require problem‐specific numerical modeling. This work extends the previous analytical solutions with more flexible boundary conditions, presents two quasianalytical methods for solving the 1‐D linear diffusive wave equation on finite domains, and applies them to different types of routing problems. Referred to as the Symbolic Diffusive Wave Solutions, the proposed methods yield explicit symbolic expressions for time‐continuous solutions at discrete nodes in space and provide solutions that are accurate and computationally efficient. The methods are easy to implement and may be used in a variety of routing applications in which accurate explicit symbolic solutions for linear advection‐diffusion are desired for a set of discrete locations such as known river forecast points. This study describes the solutions and their application in different types of real‐world and synthetic routing problems. Key Points Symbolic closed‐form solutions for 1‐D linear diffusive wave equation are developed The solutions are valid at specific nodes and are compactly expressed as explicit functions of time and channel properties The methods can handle various types of upstream inflow and downstream boundary conditions