Computing rough solutions of the stochastic nonlinear wave equation

The regularity of solutions to the stochastic nonlinear wave equation plays a critical role in the accuracy and efficiency of numerical algorithms. Rough or discontinuous initial conditions pose significant challenges, often leading to a loss of accuracy and reduced computational efficiency in existing methods. In this study, we address these challenges by developing a novel and efficient numerical algorithm specifically designed for computing rough solutions of the stochastic nonlinear wave equation, while significantly relaxing the regularity requirements on the initial data. By leveraging the intrinsic structure of the stochastic nonlinear wave equation and employing harmonic analysis tools (particularly low/high-frequency decomposition) to enable convergence analysis under low regularity, we construct a time discretization method that achieves robust convergence for initial values <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u Superscript 0 Baseline comma v Superscript 0 Baseline right-parenthesis element-of upper H Superscript gamma times upper H Superscript gamma minus 1"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> γ </mml:mi> </mml:mrow> </mml:msup> <mml:mo> × </mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> γ </mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(u^{0}, v^{0}) \in H^{\gamma } \times H^{\gamma -1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> γ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Notably, our method attains an improved error rate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis tau Superscript 2 gamma minus Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi> τ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi> γ </mml:mi> <mml:mo> − </mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(\tau ^{2\gamma -})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in one and two dimensions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma element-of left-parenthesis 0 comma one half right-bracket"> <mml:semantics> <mml:mrow> <mml:mi> γ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma \in (0, \frac {1}{2}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis tau Superscript max left-parenthesis gamma comma 2 gamma minus one half minus right-parenthesis Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi> τ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi> γ </mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mi> γ </mml:mi> <mml:mo> − </mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo> − </mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(\tau ^{\max (\gamma , 2\gamma - \frac {1}{2}-)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in three dimensions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma element-of left-parenthesis 0 comma three fourths right-bracket"> <mml:semantics> <mml:mrow> <mml:mi> γ