EXTRAPOLATION OF TYPE-1 DIAGONALLY IMPLICIT MULTISTAGE INTEGRATION METHODS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 9 (12) ◽  
pp. 10949-10955
Author(s):  
A. Jameel Kadhim ◽  
A. Gorgey
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integration Methods

scholarly journals Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 502
Author(s):  
Benjamin Zanger ◽  
Christian B. Mendl ◽  
Martin Schulz ◽  
Martin Schreiber
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Optimization Problems ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
High Order ◽  
Quantum Computers ◽  
Integration Methods ◽  
Linear Ordinary Differential Equations ◽  
Legendre Collocation Method

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

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