General Runge–Kutta–Nyström Methods for Linear Inhomogeneous Second-Order Initial Value Problems

In this paper, general Runge–Kutta–Nyström (GRKN) methods are developed and analyzed, tailored for second-order initial value problems of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mrow><mo>″</mo></mrow></msup><mo>=</mo><mi>L</mi><msup><mi>y</mi><mo>′</mo></msup><mo>+</mo><mi>M</mi><mi>y</mi><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>,</mo><mi>M</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula> are constant matrices with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The construction of embedded pairs of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>6</mn><mo>(</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>7</mn><mo>(</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>, suitable for adaptive integration strategies, is emphasized. By utilizing rooted tree theory and recent simplifications for linear inhomogeneous systems, symbolic order conditions are derived, and efficient schemes are designed through algebraic and evolutionary techniques. Numerical tests verify the superiority of our new derived pairs. In particular, this work introduces novel embedded GRKN pairs with reduced-order conditions that exploit the linearity and structure of the underlying system, enabling the construction of low-stage, high-accuracy integrators. The methods incorporate FSAL (First Same As Last) formulations, making them computationally efficient. They are tested on representative physical systems in one, two, and three dimensions, demonstrating notable improvements in efficiency and accuracy over existing high-order RKN methods.