Two-dimensional Wave Research Articles

Assessment of the utility of two-dimensional shear wave elastography and superb microvascular imaging in postoperative patients with biliary atresia.

We aimed to investigate whether prediction of liver fibrosis using two-dimensional shear wave elastography (2D-SWE) and vascular tree grading using superb microvascular imaging (SMI) are useful for postoperative follow-up in patients with biliary atresia (BA). We retrospectively collected data from medical records of 134 patients who underwent ultrasound examination with 2D-SWE or SMI, including 13 postoperative patients with BA and 121 non-BA patients. We investigated the distribution of liver stiffness values with SWE and vascular tree grading with SMI and evaluated correlations between these findings and biochemical indices of liver fibrosis in postoperative BA patients. The SWE values of the BA group were not significantly different from that of any other disease groups in non-BA patients. In postoperative BA patients, SWE values correlated significantly with aspartate aminotransferase to platelet ratio index (Spearman rank correlation coefficient [rs] = 0.6380, p = 0.0256) and with the Fib-4 index (rs = 0.6526, p = 0.0214). SMI vascular tree grading of the BA group was significantly higher than that of the choledochal cyst group (p = 0.0008) and other hepatobiliary disorder group (p = 0.0030). In postoperative BA patients, SMI vascular tree grading was not positively correlated with any biochemical marker of fibrosis. 2D-SWE appears to be useful for follow-up in postoperative BA patients.

Comparison of vibration-controlled transient elastography and two-dimensional shear wave elastography for liver stiffness measurement in patients with metabolic dysfunction-associated steatotic liver disease

Comparison of vibration-controlled transient elastography and two-dimensional shear wave elastography for liver stiffness measurement in patients with metabolic dysfunction-associated steatotic liver disease

Understanding confounding factors allows for accurate interpretation of liver stiffness measurements by ElastQ, a novel 2D shear wave elastography technique.

Liver stiffness measurement (LSM) using vibration-controlled transient elastography (VCTE) or two-dimensional shear wave elastography (2D-SWE) is recommended to assess the risk of liver fibrosis and advanced chronic liver disease. Even though both techniques measure liver stiffness, their numerical results often diverge. Confounders and reliability criteria for 2D-SWE have not been systematically investigated. We prospectively recruited participants with paired LSM by VCTE and the novel 2D-SWE technique ElastQ (Philips) in three European tertiary centers. The following parameters were recorded: sex, age, body mass index (BMI), etiology, laboratory markers of liver damage and function, as well as cholestasis, LSM by VCTE and controlled attenuation parameter (CAP), interquartile range (IQR)/median for VCTE-LSM and ElastQ-LSM, and the skin-to-liver capsule distance. We included 840 participants: 447 (53.2%) males; median age 57.0 [IQR:19.0] years; median BMI 25.4 [6.0] kg/m2; median VCTE-LSM 7.25 [9.2] kPa; median ElastQ-LSM 6.7 [5.4] kPa. On uni- and multivariable modeling (adjusted for LSM), we found that the discrepancy increased with liver stiffness and markers of disease severity. Skin-to-liver capsule distance and BMI affected VCTE-LSM more compared to ElastQ-LSM and significantly increased the discordance between the two measurements. The discrepancy of ElastQ-LSM to VCTE-LSM increases with liver stiffness and disease severity. BMI and skin-to-liver capsule distance increase the discrepancy between VCTE- and ElastQ-LSM but affect ElastQ-LSM less. The quality criterion IQR/median ≤ 30% indicates reliable ElastQ-LSM.

On the Effect of Two-Dimensional Plasma Inhomogeneity in a Magnetic Island on the Parametric Excitation Threshold of Trapped Upper Hybrid Waves and the Level of Anomalous Absorption in ECRH Experiments

The effect of two-dimensional localization of an upper hybrid (UH) wave in a magnetic island is found. The influence of this effect on the threshold and saturation level of the absolute parametric decay instability of an extraordinary wave, which results in the excitation of two two-dimensional localized UH waves, is investigated.

Symmetric irrotational deep-water waves are traveling waves

In this paper, we show that a periodic solution to the irrotational two-dimensional deep-water wave problem with the horizontal velocity components at the surface, and wave profiles are symmetric, and periodic in the x-variable, necessarily defines a traveling wave. The proof relies on employing the Maximum principle for harmonic functions in unbounded domains and structural properties of the governing equations for nonlinear deep-water waves.

Specific metabolic impairments indicate loss of sustained liver improvements in metabolic dysfunction-associated steatotic liver disease treatment.

High liver fat content (LFC) induces increased risks of both hepatic and extrahepatic progression in metabolic dysfunction-associated steatotic liver disease (MASLD), while maintaining a significant decline in magnetic resonance imaging-based proton density fat fraction (MRI-PDFF) (≥30% decline relative to baseline) without worsening fibrosis results in improved histological severity and prognosis. However, the factors associated with the loss of sustained responses to treatment remain unclear, and we aim to identify them. Consecutive treatment-naïve MASLD patients between January 2015 and February 2022, with follow-up until April 2023, were included in this prospective cohort study. LFC quantified by MRI-PDFF and liver stiffness measurements (LSM) determined by two-dimensional shear wave elastography (2D-SWE) were evaluated at weeks 0, 24 and 48. MRI-PDFF response was defined as a ≥30% relative decline in PDFF values, and LSM response was defined as a ≥1 stage decline from baseline. A total of 602 MASLD patients were enrolled. Of the 303 patients with a 24-week MRI-PDFF response and complete follow-up of 48 weeks, the rate of loss of MRI-PDFF response was 29.4%, and multivariable logistic regression analyses showed that 24-week insulin resistance (IR), still regular exercise and caloric restriction after 24 weeks, and the relative decline in LFC were risk factors for loss of MRI-PDFF response. Loss of LSM response at 48 weeks occurred in 15.9% of patients, and multivariable analysis confirmed 24-week serum total bile acid (TBA) levels and the relative decline in TBA from baseline as independent predictors. No significant association was found at 48 weeks between loss of MRI-PDFF response and loss of LSM response. MASLD patients with IR and high TBA levels are at higher risks of subsequent diminished sustained improvements of steatosis and fibrosis, respectively.

Evaluating fetal lung development at various gestational weeks using two-dimensional shear wave elastography.

Noninvasive evaluation of fetal lung development is a critical area of study. Two-dimensional shear-wave elastography (2D-SWE) provides valuable insights into tissue stiffness, potentially correlating with different stages of lung development. This study aims to explore the potential of the 2D-SWE technique for assessing the maturity of fetal lung development. This prospective cohort study included pregnant women undergoing routine antenatal ultrasound examinations at the Second Affiliated Hospital of Fujian Medical University and Quanzhou Women's and Children's Hospital from September 2022 to September 2023. The study consecutively recruited 300 pregnant women with normal pregnancies and 15 who opted for induced labor. Among those with normal pregnancies, the study assessed the differences in fetal pulmonary and hepatic elasticity measurements across different gestational weeks (GW) using one-way analysis of variance (ANOVA). Furthermore, regression analyses using linear, quadratic, and cubic equations were conducted to investigate the relationship between fetal parameters and GW. For those who opted for induced labor, elasticity measurements were taken before induction, and fetal lung tissue specimens were collected for post-induction observation. Fetal lung and liver elasticity values, along with the lung-to-liver elasticity ratio (LLE ratio), showed significant variations across different GW (P<0.05). Specifically, fetal lung elasticity values initially increased and then decreased as GW advanced (R2=0.41). Liver elasticity values continuously increased throughout GW, though the rate of increase diminished during the prenatal period (R2=0.37). The LLE ratio values increased and then decreased over GW, fluctuating overall between 0.8 and 0.9 (R2=0.14). A 71.4% concordance was observed between the predicted stage of lung development, based on lung elasticity values, and the histological stage of lung development in the induced fetuses. 2D-SWE can depict the maturation of fetal lung development at various stages.

Numerical modeling of wave interaction with a porous floating structure consisting of uniform spheres

Porous materials are increasingly being used in the design of floating structures in coastal and ocean engineering, but there is a lack of numerical tools that can aid in the design of a movable floating porous structure. To close this gap, the existing volume-averaged numerical model for flow interacting with a fixed porous body was extended to floating scenarios by (1) using the relative velocity in the porous friction force, (2) calculating rigid body motion using volume integral of porous body force, and (3) modifying a dynamic mesh algorithm for a mobile porous body. As a demonstration, the developed model was applied to a porous floating structure consisting of cubically packed uniform spheres. Two sets of model applications were involved. The first set considered three-dimensional flow around a fixed porous block placed beneath the free surface. The measured total force on the block under wave or steady flows was predicted accurately with an error less than 10%. The second set involved a two-dimensional wave interacting with a floating porous block representing a breakwater. For free-floating conditions, the model can accurately predict the dynamic response of the structure, including the time varying movement of its rigid body and the mean drift. For mooring-restrained conditions, the mooring force and wave transmission coefficients were also predicted well with an error less than 20%. The proposed numerical approach can be applied to other floating structures with a rigid volumetric porous body. Future research is also required to study the microscopic pore flows, upon which more detailed parameterization of the porous media can be derived.

A variable kinematic multi-field model for lamb wave propagation analysis in smart composite panels

The study explores the propagation of symmetric and antisymmetric fundamental Lamb waves in laminated composite strips, employing multi-field piezo-elastic plate models. The findings indicate the necessity of employing higher-order kinematic models to enhance the accuracy of wave propagation assessments. To further optimize computational efficiency, node-dependent kinematic models are utilized. The research delves into the tradeoff between reducing computational costs and maintaining result accuracy demonstrating a reduction of up to 60% in computational expenses in simple models while keeping the error below 1%. Two-dimensional wave propagation in laminated composite plates was further considered, expanding the scope due to material orthotropy.

Authentic fault models and dispersive tsunami simulations for outer-rise normal earthquakes in the southern Kuril Trench

The southern Kuril Trench is one of the most seismically active regions in the world. In this study, marine surveys and observations were performed to construct fault models for possible outer-rise earthquakes. Seismic and seafloor bathymetric surveys indicated that the dip angle of the outer-rise fault was approximately 50°–80°, with a strike that was slightly oblique to the axis of the Kuril Trench. The maximum fault length was estimated to be ~ 260 km. Based on these findings, we proposed 17 fault models, with moment magnitudes ranging from 7.2 to 8.4. To numerically simulate tsunami, we solved two-dimensional dispersive wave and three-dimensional Euler equations using the outer-rise fault models. The results of both simulations yielded identical predictions for tsunami with short-wavelength components, resulting in significant dispersive deformations in the open ocean. We also found that tsunami generated by outer-rise earthquakes were affected by refraction and diffraction because of the source location beyond the trench axis. These findings can improve future predictions of tsunami hazards.Graphical

Liver stiffness in hepatocellular carcinoma and chronic hepatitis patients: Hepatitis B virus infection and transaminases should be considered.

Liver condition is a crucial prognostic factor for patients with hepatocellular carcinoma (HCC), but a convenient and comprehensive method to assess liver condition is lacking. Liver stiffness (LS) measured by two-dimensional shear wave elastography may help in assessing liver fibrosis and liver condition. Chronic hepatitis B (CHB) is an important risk factor for HCC progression, but LS was found to be less reliable in assessing liver fibrosis following hepatitis viral eradication. We hypothesize that the status of hepatitis virus infection would affect the accuracy of LS in assessing the liver condition. To test the feasibility and impact factors of using LS to assess liver condition in patients with HCC and CHB. A total of 284 patients were retrospectively recruited and classified into two groups on the basis of serum CHB virus hepatitis B virus (HBV)-DNA levels [HBV-DNA ≥ 100.00 IU/mL as Pos group (n = 200) and < 100.00 IU/mL as Neg group (n = 84)]. Correlation analyses and receiver operating characteristic analyses were conducted to evaluate the relationship between LS and liver condition. A significant correlation was found between LS and most of the parameters considered to have the ability to evaluate liver condition (P < 0.05). When alanine aminotransferase (ALT) concentrations were normal (≤ 40 U/L), LS was correlated with liver condition indices (P < 0.05), but the optimal cutoff of LS to identify a Child-Pugh score of 5 was higher in the Neg group (9.30 kPa) than the Pos group (7.40 kPa). When ALT levels were elevated (> 40 U/L), the correlations between LS and liver condition indices were not significant (P > 0.05). LS was significantly correlated with most liver condition indices in patients with CHB and HCC. However, these correlations varied according to differences in HBV-DNA and transaminase concentrations.

Nomogram based on liver stiffness and spleen area with ultrasound for posthepatectomy liver failure: A multicenter study.

Liver stiffness (LS) measurement with two-dimensional shear wave elastography (2D-SWE) correlates with the degree of liver fibrosis and thus indirectly reflects liver function reserve. The size of the spleen increases due to tissue proliferation, fibrosis, and portal vein congestion, which can indirectly reflect the situation of liver fibrosis/cirrhosis. It was reported that the size of the spleen was related to posthepatectomy liver failure (PHLF). So far, there has been no study combining 2D-SWE measurements of LS with spleen size to predict PHLF. This prospective study aimed to investigate the utility of 2D-SWE assessing LS and spleen area (SPA) for the prediction of PHLF in hepatocellular carcinoma (HCC) patients and to develop a risk prediction model. To investigate the utility of 2D-SWE assessing LS and SPA for the prediction of PHLF in HCC patients and to develop a risk prediction model. This was a multicenter observational study prospectively analyzing patients who underwent hepatectomy from October 2020 to March 2022. Within 1 wk before partial hepatectomy, ultrasound examination was performed to measure LS and SPA, and blood was drawn to evaluate the patient's liver function and other conditions. Least absolute shrinkage and selection operator logistic regression and multivariate logistic regression analysis was applied to identify independent predictors of PHLF and develop a nomogram. Nomogram performance was validated further. The diagnostic performance of the nomogram was evaluated with receiver operating characteristic curve compared with the conventional models, including the model for end-stage liver disease (MELD) score and the albumin-bilirubin (ALBI) score. A total of 562 HCC patients undergoing hepatectomy (500 in the training cohort and 62 in the validation cohort) were enrolled in this study. The independent predictors of PHLF were LS, SPA, range of resection, blood loss, international normalized ratio, and total bilirubin. Better diagnostic performance of the nomogram was obtained in the training [area under receiver operating characteristic curve (AUC): 0.833; 95% confidence interval (95%CI): 0.792-0.873; sensitivity: 83.1%; specificity: 73.5%] and validation (AUC: 0.802; 95%CI: 0.684-0.920; sensitivity: 95.5%; specificity: 52.5%) cohorts compared with the MELD score and the ALBI score. This PHLF nomogram, mainly based on LS by 2D-SWE and SPA, was useful in predicting PHLF in HCC patients and presented better than MELD score and ALBI score.

Potential role of two-dimensional shear wave elastography, including liver stiffness measurement and dispersion slope, for management of chronic hepatitis B.

This study assessed the use of liver stiffness (LS) and dispersion slope (DS) in classifying patients with chronic hepatitis B (CHB) by the necessity of antiviral treatment. A retrospective review examined 249 patients with CHB (male:female, 107:142; mean age, 53±14 years) between April 2018 and March 2022. Patients for whom treatment was indicated, termed group 1, exhibited either serum alanine transaminase (ALT) and aspartate aminotransferase levels not exceeding five times the upper limit of normal and LS &gt;13 kPa, or hepatitis B virus DNA &gt;2,000 IU/mL and ALT &gt;40 IU/L. The remaining patients comprised group 2 (treatment not required). In subgroup analysis, patients with LS &lt;13 kPa were further examined. Overall, group 1 exhibited significantly higher LS (12.64±6.76 vs. 6.31±1.67 kPa, P&lt;0.001) and DS (15.52±4.95 vs. 11.04±1.93 [m/s]/kHz, P&lt;0.001) than group 2. In subgroup analysis, group 1 also demonstrated significantly higher values (LS: 8.05±2.12 vs 6.31±1.67 kPa, P&lt;0.001, DS: 13.06±2.75 vs. 11.04±1.93 [m/s]/kHz, P&lt;0.001). The areas under the curve (AUCs) for LS and DS in group 1 were 0.855 (95% confidence interval [CI], 0.80 to 0.90; P&lt;0.001) and 0.810 (95% CI, 0.75 to 0.86; P&lt;0.001), respectively. In subgroup analysis, the AUCs for LS and DS in group 1 were 0.751 (95% CI, 0.69 to 0.81; P&lt;0.001) and 0.711 (95% CI, 0.64 to 0.77; P&lt;0.001), respectively. Within group 1, the AUCs for LS and DS did not differ significantly (P&gt;0.05). LS and DS assist in classifying patients with CHB by their need for antiviral treatment.

Enhancing flood wave modelling of reservoir failure: a comparative study of structure-from-motion based 2D and 3D methodologies

Predicting flood wave propagation from reservoir failures is critical to practical flood hazard assessment and risk management. Flood waves are sensitive to topography, channel geometry, structures, and natural features along floodplain paths. Thus, the accuracy of flood wave modelling depends on how precisely those features are represented. This study introduces an enhancing approach to flood wave modelling by accurately representing three-dimensional objects in floodplains using the structure-from-motion (SfM). This method uses an unmanned aerial vehicle to capture topographic complexities and account for ground objects that impact flood propagation. Using the three-dimensional volume of fluid numerical approach significantly improves an enhanced representation of turbulent flow dynamics and computational efficiency, especially in handling large topography datasets. Reproductions from this enhanced three-dimensional approach were validated against recent reservoir failure observations and contrasted with traditional two-dimensional models. The results revealed that the suggested three-dimensional methodology achieved a significant 84.4% reproducibility when juxtaposed with actual inundation traces. It was 35.5%p more accurate than the two-dimensional diffusion wave equation (DWE) and 17.1%p more than the shallow water equation (SWE) methods in predicting flood waves. This suggests that the reproducibility of the DWE and SWE decreases compared to the three-dimensional approach when considering more complex floodplains. These results demonstrate that three-dimensional flood wave analysis with the SfM methodology is optimal for effectively minimising topographic and flood wave reproduction errors across extensive areas. This dual reduction in errors significantly enhances the reliability of flood hazard assessments and improves risk management by providing more precise and realistic predictions of flood waves.

Volumetric Measurement Of The Acoustically Induced Fluid Flow In A Micro Chamber With A Two-Dimensional Standing Surface Acoustic Wave Field

The non-invasive, contactless and label-free trapping and patterning of individual cells with acoustic tweezers is a key technology to reliably analyze single cells and cell-cell interactions in biological applications. By using two-dimensional standing surface acoustic waves (2DsSAW), the precise manipulation of the cells in a micro chamber is primarily governed by the acoustic radiation force, which acts directly on the cells due to scattering effects of the acoustic waves. However, a fluid flow of three-dimensional nature is concurrently induced acoustically, which additionally influences the cell pattern by Stokes' drag. In this study, we shed light on the complex acoustically induced fluid flow forming in an acoustic tweezer by 3D3C velocity measurements using the astigmatism particle tracking velocimetry (APTV). Vortex structures above the antinodes of the 2DsSAW field were revealed. The fluid flow converge and diverge in the vicinity of the chamber bottom and ceiling, respectively. Hence, the fluid flow influences the stable trapping locations of cells or particles in different ways depending on their height position, which might explain the recently reported formation of various particle patterns at different height levels. Moreover, the experimental findings are used to validate a three-dimensional numerical model. The numerical model provides deep insights into the underlying acoustic fields responsible for the formation of the fluid flow and acoustic radiation force. By combining the experimental and numerical results, a profound understanding of the physical mechanisms for cell or particle trapping and patterning is established.

Quantitative diagnosis of early acute compartment syndrome using two-dimensional shear wave elastography in a rabbit model.

This study explored the association of the elasticity modulus and shear wave velocity (SWV) of the tibialis anterior muscle, as measured by two-dimensional shear wave elastography (2D-SWE), with the intracompartmental pressure (ICP) determined using the Whitesides method in a New Zealand rabbit model of acute compartment syndrome (ACS). Additionally, it evaluated the viability of 2D-SWE as a noninvasive, quantitative tool for the early detection of ACS. An ACS model was established through direct external compression by applying pressure bandaging to the lower legs of 15 New Zealand rabbits using neonatal blood pressure cuffs. Another five animals represented a non-modeled control group. To measure the elasticity modulus and SWV of the tibialis anterior muscles, 2D-SWE was employed. Blood oxygen saturation, serum creatine kinase (CK), and myoglobin levels were monitored. Subsequently, the anterior tibial compartment was dissected, and the tibialis anterior was removed for hematoxylin and eosin staining to assess muscle injury. The elasticity modulus and SWV of the tibialis anterior muscle increased with compression duration, as did serum CK and myoglobin levels. ICP was strongly positively correlated with these parameters, particularly mean velocity (r=0.942, P&lt;0.001) and CK (r=0.942, P&lt;0.001). Blood oxygen saturation was negatively correlated with ICP (r=-0.887, P&lt;0.001). Histological analysis indicated progressive muscle cell swelling over time, with damage transitioning from reversible to irreversible and culminating in necrosis. In a rabbit ACS model, ICP was strongly positively correlated with muscle elasticity modulus/SWV. Consequently, 2D-SWE may represent a novel tool for assessing early-phase ACS.

Examining basic theorems and plane waves in the context of thermoelastic diffusion using a multi-phase-lag model with temperature dependence

This study employs a novel mathematical formulation of temperature-dependent thermoelastic diffusion with multi-phase delays (TDMT). The TDMT model, which is assumed, is obtained by converting the fundamental rules of Fourier’s and Fick’s laws into higher order time derivatives of the heat flow vector, the temperature gradient, diffusing mass flux, and the chemical potential gradient. Fundamental theorems such as energy, uniqueness, reciprocity theorems, and variational criterion are established using the fundamental equations for the assumed model TDMT. The reciprocity theorem is applied to a particular scenario by using instantaneous concentrated body forces, heat sources, chemical potential sources, and moving heat sources as examples. It is noted that both the variational criterion and these theorems depend on the field variables’ susceptibility as well as the fluctuations in the higher order temporal derivatives’ parameters. Additionally, the assumed model’s two-dimensional (2D) plane wave propagation is described. Three longitudinal waves are identified: the longitudinal (P) wave, the thermal (T) wave, the chemical potential (CP) wave, and the linked transverse (SV) wave. Wave properties (phase velocity and attenuation coefficient) are calculated numerically and shown visually. A few special examples are also investigated and compared with the established findings. The framework for additional research into basic issues in thermoelastic continua under various physical field variables is provided by this work. Numerous applications in material science, geomechanics, soil dynamics, and the electronic sector can be made of the current findings.

Diagnostic performance of two-dimensional shear wave elastography and attenuation imaging for fibrosis and steatosis assessment in chronic liver disease.

We investigated the diagnostic performance of two-dimensional shear wave elastography (2D-SWE) and attenuation imaging (ATI) in detecting fibrosis and steatosis in patients with chronic liver disease (CLD), comparing them with established methods. In 190 patients with CLD, 2D-SWE and vibration-controlled transient elastography (VCTE) were used for liver stiffness measurement (LSM), and ATI and controlled attenuation parameter (CAP) were used for steatosis quantification. The correlations between these new and established methods were analyzed. Significant correlations were found between 2D-SWE and VCTE (r = 0.78, P < 0.001), and between ATI and CAP (r = 0.70, P < 0.001). Liver stiffness tended to be lower with 2D-SWE compared with that with VCTE, especially in cases with higher LSM, and ATI was less influenced by skin-capsular distance than CAP. Area under the receiver-operating characteristics curves (AUCs) and optimal cut-offs of 2D-SWE for diagnosing liver fibrosis stages F2, F3, and F4 were 0.73 (8.7kPa), 0.79 (9.1kPa), and 0.88 (11.6kPa), respectively. The AUCs and optimal cut-offs of ATI for diagnosing hepatic steatosis grades S1, S2, and S3 were 0.91 (0.66dB/cm/MHz), 0.80 (0.79dB/cm/MHz), and 0.88 (0.86dB/cm/MHz), respectively. A subgroup analysis of 86 patients with metabolic dysfunction-associated steatotic liver disease also demonstrated good performance for 2D-SWE and ATI. 2D-SWE and ATI performed comparably with conventional VCTE and CAP in evaluating CLD, offering reliable alternatives for diagnosing liver fibrosis and steatosis.

Embedded exponential Runge–Kutta–Nyström methods for highly oscillatory Hamiltonian systems

Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from O(N2) to O(Nlog⁡N). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs.

Wave propagation on hexagonal lattices

Abstract We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem.