TY - JOUR

T1 - Enhancing the order of the Milstein scheme for stochastic partial differential equations with commutative noise

AU - Leonhard, Claudine

AU - Roßler, Andreas

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We consider a higher-order Milstein scheme for stochastic partial differential equations (SPDEs) with trace class noise which fulfill a certain commutativity condition. A novel technique to generally improve the order of convergence of Taylor schemes for SPDEs is introduced. The key tool is an efficient approximation of the Milstein term by particularly tailored nested derivative-free terms. For the resulting derivative-free Milstein scheme the computational cost is, in general, considerably reduced by some power. Further, a rigorous computational cost model is considered, and the so-called effective order of convergence is introduced, which allows direct comparison of various numerical schemes in terms of their efficiency. As the main result, we prove for a broad class of SPDEs, including equations with operators that do not need to be pointwise multiplicative, that the effective order of convergence of the proposed derivative-free Milstein scheme is significantly higher than for the original Milstein scheme. In this case, the derivative-free Milstein scheme outperforms the Euler scheme as well as the original Milstein scheme due to the reduction of the computational cost. Finally, we present some numerical examples that confirm the theoretical results.

AB - We consider a higher-order Milstein scheme for stochastic partial differential equations (SPDEs) with trace class noise which fulfill a certain commutativity condition. A novel technique to generally improve the order of convergence of Taylor schemes for SPDEs is introduced. The key tool is an efficient approximation of the Milstein term by particularly tailored nested derivative-free terms. For the resulting derivative-free Milstein scheme the computational cost is, in general, considerably reduced by some power. Further, a rigorous computational cost model is considered, and the so-called effective order of convergence is introduced, which allows direct comparison of various numerical schemes in terms of their efficiency. As the main result, we prove for a broad class of SPDEs, including equations with operators that do not need to be pointwise multiplicative, that the effective order of convergence of the proposed derivative-free Milstein scheme is significantly higher than for the original Milstein scheme. In this case, the derivative-free Milstein scheme outperforms the Euler scheme as well as the original Milstein scheme due to the reduction of the computational cost. Finally, we present some numerical examples that confirm the theoretical results.

UR - http://www.scopus.com/inward/record.url?scp=85053545991&partnerID=8YFLogxK

U2 - 10.1137/16M1094087

DO - 10.1137/16M1094087

M3 - Journal articles

AN - SCOPUS:85053545991

SN - 0036-1429

VL - 56

SP - 2585

EP - 2622

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 4

ER -