Very High-Order A-stable Stiffly Accurate Diagonally Implicit Runge-Kutta Methods with Error Estimators

  • Yousef Alamri

Student thesis: Master's Thesis


A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of which have identical diagonal elements (SDIRK) and explicit first stage (ESDIRK). In each of these classes, we present new A-stable schemes of orders six (the highest order of previously known A-stable DIRK-type schemes) up to order eight. For each order, we include one scheme that is only A-stable as well as one that is stiffly accurate and/or L-stable. The latter require more stages but give better results for highly stiff problems and differential-algebraic equations (DAEs). The development of the eighth-order schemes requires, in addition to imposing A-stability, finding highly accurate numerical solutions for a system of 200 equations in over 100 variables, which is accomplished via a combination of global and local optimization strategies. The accuracy, stability, and efficiency of the schemes are demonstrated on diverse problems.
Date of AwardJul 24 2023
Original languageEnglish (US)
Awarding Institution
  • Computer, Electrical and Mathematical Sciences and Engineering
SupervisorDavid Ketcheson (Supervisor)


  • Runge-Kutta Methods
  • Time Integration
  • Numerical Analysis
  • Differential Equations

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