<p>In the nonrelativistic limit regime, the massive Klein-Gordon-Dirac (KGD) system involves a small dimensionless parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<\varepsilon \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ε</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and admits rapid oscillations with a wavelength <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(\varepsilon ^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in time as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. Meanwhile, due to the coupled Yukawa interaction term, the fundamental design of the existing multiscale time integrator discretization presents a significant barrier to higher-order convergence. A fundamentally different strategy is required to transcend this order barrier. In this paper, we consider arbitrary high-order discretizations of the KGD system in the nonrelativistic limit regime by applying the nested Picard iterative integrators (NPIs) based on the Picard iteration with optimal and uniform accuracy w.r.t. <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> to effectively decouple the challenging Yukawa interaction terms. This approach is valid to design efficient and explicit numerical schemes, which are easy to implement and perform significantly better than the classical ones in the literature for the KGD system in the nonrelativistic limit regime. More specifically, we rigorously establish the optimal and uniform error bounds of the arbitrary high-order NPI discretizations of the KGD system in the nonrelativistic limit regime. Finally, extensive numerical results demonstrate that the error bounds in this paper are optimal and sharp.</p>

Uniformly Accurate Nested Picard Iterative Integrators for The Klein-Gordon-Dirac System in The Nonrelativistic Limit Regime

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发表日期：

2026年7月31日

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Yongyong Cai,
Wenfan Yi

摘要

In the nonrelativistic limit regime, the massive Klein-Gordon-Dirac (KGD) system involves a small dimensionless parameter $0<\varepsilon \le 1$ $0 < ε \leq 1$ and admits rapid oscillations with a wavelength $O(\varepsilon ^2)$ $O (ε^{2})$ in time as $\varepsilon \rightarrow 0^+$ $ε \to 0^{+}$ . Meanwhile, due to the coupled Yukawa interaction term, the fundamental design of the existing multiscale time integrator discretization presents a significant barrier to higher-order convergence. A fundamentally different strategy is required to transcend this order barrier. In this paper, we consider arbitrary high-order discretizations of the KGD system in the nonrelativistic limit regime by applying the nested Picard iterative integrators (NPIs) based on the Picard iteration with optimal and uniform accuracy w.r.t. $\varepsilon $ $ε$ to effectively decouple the challenging Yukawa interaction terms. This approach is valid to design efficient and explicit numerical schemes, which are easy to implement and perform significantly better than the classical ones in the literature for the KGD system in the nonrelativistic limit regime. More specifically, we rigorously establish the optimal and uniform error bounds of the arbitrary high-order NPI discretizations of the KGD system in the nonrelativistic limit regime. Finally, extensive numerical results demonstrate that the error bounds in this paper are optimal and sharp.