Higher-order Runge--Kutta type schemes for the Method of Characteristics
Conference Year
January 2019
Abstract
The Method of Characteristics (MoC) is a well-known and popular method used to find the numerical solution of systems of hyperbolic partial differential equations (PDEs). The main idea of the MoC is to solve a system of ordinary differential equations (ODEs) along the characteristic curves admitted by the PDEs. In principle, this can be accomplished by using any appropriate numerical method for ODEs. We examine the case of hyperbolic systems which have straight-line and crossing characteristics, for which only second-order accurate MoC schemes have been reported. The purpose of our research is to develop numerically stable MoC schemes that are third- and fourth-order accurate. We show that the desired accuracy can be obtained by using the classical third- and fourth-order Runge—Kutta (RK) schemes as the ODE solver in the MoC, but that the resulting methods are numerically unstable for certain systems of PDEs. To achieve numerical stability as well as the desired accuracy, we employ the so-called pseudo-RK (pRK) methods as the ODE solver, which are at the intersection of RK methods and multistep methods. We then explore the ability of MoC-pRK schemes to preserve some of the conservation laws commonly admitted by hyperbolic systems.
Primary Faculty Mentor Name
Taras Lakoba
Faculty/Staff Collaborators
Taras Lakoba (Advisor)
Status
Graduate
Student College
College of Engineering and Mathematical Sciences
Program/Major
Mathematics
Primary Research Category
Engineering & Physical Sciences
Higher-order Runge--Kutta type schemes for the Method of Characteristics
The Method of Characteristics (MoC) is a well-known and popular method used to find the numerical solution of systems of hyperbolic partial differential equations (PDEs). The main idea of the MoC is to solve a system of ordinary differential equations (ODEs) along the characteristic curves admitted by the PDEs. In principle, this can be accomplished by using any appropriate numerical method for ODEs. We examine the case of hyperbolic systems which have straight-line and crossing characteristics, for which only second-order accurate MoC schemes have been reported. The purpose of our research is to develop numerically stable MoC schemes that are third- and fourth-order accurate. We show that the desired accuracy can be obtained by using the classical third- and fourth-order Runge—Kutta (RK) schemes as the ODE solver in the MoC, but that the resulting methods are numerically unstable for certain systems of PDEs. To achieve numerical stability as well as the desired accuracy, we employ the so-called pseudo-RK (pRK) methods as the ODE solver, which are at the intersection of RK methods and multistep methods. We then explore the ability of MoC-pRK schemes to preserve some of the conservation laws commonly admitted by hyperbolic systems.