While martensites subjected to quasi-static deformation are known to exhibit power law distributed acoustic emission in a broad range of scales the origin of the observed scaling behavior and the mechanism for self-organization towards criticality remains obscure. Here, we argue that the power-law structure of intermittent fluctuations can be at least partially attributed to inertia. We build on the insight that inertial dynamics, evidenced by acoustic emission, can become an important factor if the underlying mechanical system is only marginally stable. We first illustrate the possibility of inertia-induced heavy-tailed avalanche size distributions using a prototypical example of a discrete chain with bi-stable springs. We then explore the effects of inertia in fully realistic two- and three-dimensional continuum models of elastic phase transitions. In particular, we demonstrate that a three-dimensional model of this type can produce not only qualitative but also quantitative agreement with experiment.
PlanesAVivesE. Avalanche criticality in thermal-driven martensitic transitions: the asymmetry of the forward and reverse transitions in shape-memory materials. J Phys Condens Matter2017; 29(33): 334001.
2.
PortaMCastánTSaxenaA, et al. Influence of the number of orientational domains on avalanche criticality in ferroelastic transitions. Phys Rev E2019; 100(6): 062115.
Perez-RecheFJ. Modelling avalanches in martensites. In: PlanesASaxenaASaljeEKH (eds) Avalanches in functional materials and geophysics. Cham: Springer, 2017, pp. 99–136.
5.
BalandraudXBarreraNBiscariP, et al. Strain intermittency in shape-memory alloys. Phys Rev B2015; 91(17): 174111.
6.
SethnaJPBierbaumMKDahmenKA, et al. Deformation of crystals: connections with statistical physics. Annu Rev Mater Res2017; 47(1): 217–246.
7.
AlavaMJNukalaPKZapperiS. Statistical models of fracture. Adv Phys2006; 55(3–4): 349–476.
ZhangPSalmanOUWeissJ, et al. Variety of scaling behaviors in nanocrystalline plasticity. Phys Rev E2020; 102(2–1): 023006.
14.
SalmanOUFinelADelvilleR, et al. The role of phase compatibility in martensite. J Appl Phys2012; 111(10): 103517.
15.
FisherDS. Sliding charge-density waves as a dynamic critical phenomenon. Phys Rev B1985; 31: 1396.
16.
RossoASethnaJPWyartM. Avalanches and deformation in glasses and disordered systems. In: Emergent dynamics in glasses and disordered systems: correlations and avalanches, Spin Glass Theory and Far Beyond: Replica Symmetry Breaking After 40 Years. Singapore: World Scientific Publishing, 2022, pp. 277–305.
DenisovDLorinczKUhlJ, et al. Universality of slip avalanches in flowing granular matter. Nat Commun2016; 7: 10641.
19.
LiuCFerreroEEPuosiF, et al. Driving rate dependence of avalanche statistics and shapes at the yielding transition. Phys Rev Lett2016; 116: 065501.
20.
ChanPYTsekenisGDantzigJ, et al. Plasticity and dislocation dynamics in a phase field crystal model. Phys Rev Lett2010; 105: 015502.
21.
FriedmanNJenningsATTsekenisG, et al. Statistics of dislocation slip avalanches in nanosized single crystals show tuned critical behavior. Phys Rev Lett2012; 109: 095507.
22.
BakPTangCWiesenfeldK. Self-organized criticality: an explanation of the 1/f noise. Phys Rev Lett1987; 59(4): 381–384.
23.
DickmanRMuñozMAVespignaniA, et al. Paths to self-organized criticality. Braz J Phys2000; 30: 27–41.
24.
JensenHJ. Self-organized criticality: emergent complex behavior in physical and biological systems, vol. 10. Cambridge: Cambridge University Press, 1998.
25.
DharD. Theoretical studies of self-organized criticality. Physica A2006; 369(1): 29–70.
26.
PruessnerG. Self-organised criticality: theory, models and characterisation. Cambridge: Cambridge University Press, 2012.
27.
SornetteD. Critical phenomena in natural sciences: chaos, fractals, self-organization and disorder: concepts and tools. Berlin and Heidelberg: Springer, 2006.
28.
GrosC. Self-organized criticality. In: GrosC (ed.) Complex and adaptive dynamical systems: a comprehensive introduction. Cham: Springer, 2024, pp. 203–239.
29.
TadićBMelnikR. Self-organised critical dynamics as a key to fundamental features of complexity in physical, biological, and social networks. Dynamics2021; 1(2): 181–197.
30.
MüllerMWyartM. Marginal stability in structural, Spin, and electron glasses. Annu Rev Condens Matter Phys2015; 6: 177.
RoitburdA. Martensitic transformation as a typical phase transformation in solids. Solid State phys1978; 33: 317–390.
36.
JamesRDHaneKF. Martensitic transformations and shape-memory materials. Acta Mater2000; 48(1): 197–222.
37.
KhachaturyanAG. Theory of structural transformations in solids. North Chelmsford, MA: Courier Corporation, 2013.
38.
BhattacharyaK. Microstructure of martensite. Oxford: Oxford University Press, 2003.
39.
Pérez-RecheFJStipcichMVivesE, et al. Kinetics of martensitic transitions in Cu-Al-Mn under thermal cycling: analysis at multiple length scales. Phys Rev B2004; 69: 064101.
40.
ChandniUGhoshAVijayaHS, et al. Criticality of tuning in athermal phase transitions. Phys Rev Lett2009; 102(2): 025701.
41.
RosinbergMLVivesE. Metastability, hysteresis, avalanches, and acoustic emission: martensitic transitions in
functional materials. In: PlanesASaxenaAFukudaT, et al. (eds) Disorder and strain-induced complexity in functional materials. Berlin and Heidelberg: Springer, 2011, pp. 249–272.
42.
VivesESoto ParraDEPlanesA, et al. Acoustic emission avalanches in martensitic transitions: new perspectives for the problem of source location. Solid State Phenom2011; 172: 144–149.
43.
GallardoMCManchadoJRomeroFJ, et al. Avalanche criticality in the martensitic transition of Cu 67.64 Zn 16.71 Al 15.65 shape-memory alloy: a calorimetric and acoustic emission study. Phys Rev B2010; 81(17): 174102.
44.
BekeDLDarócziLTóthLZ, et al. Acoustic emissions during structural changes in shape memory alloys. Metals2019; 9(1): 58.
45.
IllaXWinkelmayerPVivesE. Local strain variability and force fluctuations during the martensitic transition under different driving mechanisms. Phys Rev B2015; 92(18): 184107.
46.
VivesESoto-ParraDMañosaL, et al. Driving-induced crossover in the avalanche criticality of martensitic transitions. Phys Rev B2009; 80: 180101.
47.
TorrentsGIllaXVivesE, et al. Geometrical model for martensitic phase transitions: understanding criticality and weak universality during microstructure growth. Phys Rev E2017; 95(1–1): 013001.
48.
BekeDLBolgárMKTóthLZ, et al. On the asymmetry of the forward and reverse martensitic transformations in shape memory alloys. J Alloys Compd2018; 741: 106–115.
49.
SongYChenXDabadeV, et al. Enhanced reversibility and unusual microstructure of a phase-transforming material. Nature2013; 502(7469): 85–88.
VivesEPlanesA. Hysteresis and avalanches in the random anisotropy ising model. Phys Rev B2001; 63(13): 134431.
57.
Pérez-RecheFJTruskinovskyLZanzottoG. Driving-induced crossover: from classical criticality to self-organized criticality. Phys Rev Lett2008; 101(23): 27.
58.
SethnaJPDahmenKAPerkovicO. Random-field ising models of hysteresis. New York: Academic Press, 2006.
59.
Pérez-RecheFJTruskinovskyLZanzottoG. Training-induced criticality in martensites. Phys Rev Lett2007; 99(7): 075501.
60.
Perez-RecheFJTrigueroCZanzottoG, et al. Origin of scale-free intermittency in structural first-order phase transitions. Phys Rev B2016; 94(14): 144102.
61.
Perez-RecheFJTruskinovskyLZanzottoG. Martensitic transformations: from continuum mechanics to spin models and automata. Contin Mech Thermodyn2009; 21: 17–26.
62.
CarrilloLMañosaLOrtínJ, et al. Experimental evidence for universality of acoustic emission avalanche distributions during structural transitions. Phys Rev Lett1998; 81(9): 1889–1892.
63.
BonnotERomeroRMañosaL, et al. Elastocaloric effect associated with the martensitic transition in shape-memory alloys. Phys Rev Lett2008; 100(12): 125901.
64.
EshelbyJD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lon A1957; 241: 376.
65.
PicardGAjdariALequeuxF, et al. Elastic consequences of a single plastic event: a step towards the microscopic modeling of the flow of yield stress fluids. Eur J E2004; 15: 371.
66.
RossiSBiroliGOzawaM, et al. Far-from-equilibrium criticality in the random-field ising model with Eshelby interactions. Phys Rev B2023; 108(22): L220202.
67.
ClappP. How would we recognize a martensitic transformation if it bumped into us on a dark & austy night?J Phys IV1995; 5(C8): C8–11.
68.
MiaoYVlassakJJ. Explosive martensitic transformation of supercooled austenite in CuZr-based thin-film shape memory alloys. Acta Mater2020; 200: 162–170.
69.
SchwabeSLünserKSchmidtD, et al. What is the speed limit of martensitic transformations?Sci Technol Adv Mater2022; 23(1): 633–641.
70.
LinJPenceTJ. Pulse attenuation by kinetically active phase boundary scattering during displacive phase transformations. J Mech Phys Solids1998; 46(7): 1183–1211.
71.
TruskinovskyL. Nucleation and growth in elastodynamics. In: DuxburyPMPenceTJ (eds) Dynamics of crystal surfaces and interfaces. Boston, MA: Springer, 2002, pp. 185–197.
72.
LookmanTShenoySRRasmussenKO, et al. Ferroelastic dynamics and strain compatibility. Phys Rev B2003; 67(2): 024114.
73.
TruskinovskyLVainchteinA. Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia. Contin Mech Thermodyn2008; 20(2): 97–122.
74.
SteinbachIShchygloO. Phase-field modelling of microstructure evolution in solids: perspectives and challenges. Curr Opin Solid State Mater Sci2011; 15(3): 87–92.
75.
ChoJYIdesmanALevitasV, et al. Finite element simulations of dynamics of multivariant martensitic phase transitions based on Ginzburg–Landau theory. Int J Solids Struct2012; 49(14): 1973–1992.
76.
LiuXSchneiderDRederM, et al. Modeling of martensitic phase transformation accounting for inertia effects. Int J Mech Sci2024; 278: 109443.
77.
VivesEOrtínJMañosaL, et al. Distributions of avalanches in martensitic transformations. Phys Rev Lett1994; 72(11): 1694–1697.
78.
MañosaLPlanesARoubyD, et al. Acoustic emission field during thermoelastic martensitic transformations. Appl Phys Lett1989; 54(25): 2574–2576.
79.
BaróJCorralÁIllaX, et al. Statistical similarity between the compression of a porous material and earthquakes. Phys Rev Lett2013; 110(8): 088702.
80.
GombergJJohnsonP. Dynamic triggering of earthquakes. Nature2005; 437(7060): 830–830.
81.
JohnsonPAJiaX. Nonlinear dynamics, granular media and dynamic earthquake triggering. Nature2005; 437(7060): 871–874.
82.
KocharyanG. Nucleation and evolution of sliding in continental fault zones under the action of natural and man-made factors: a state-of-the-art review. Izv Phys Solid Earth2021; 57: 439–473.
83.
KhfifiMLoulidiM. Scaling properties of a rice-pile model: inertia and friction effects. Phys Rev E2008; 78: 051117.
84.
MaimonRSchwarzJM. Continuous depinning transition with an unusual hysteresis effect. Phys Rev Lett2004; 92: 255502.
85.
PapanikolaouS. Shearing a glass and the role of pinning delay in models of interface depinning. Phys Rev E2016; 93: 032610.
86.
CarlsonJMLangerJS. Properties of earthquakes generated by fault dynamics. Phys Rev Lett1989; 62: 2632.
87.
PradoCPCOlamiZ. Inertia and break of self-organized criticality in sandpile cellular-automata models. Phys Rev A1992; 45: 665.
88.
HeldGASolinaDHKeaneDT, et al. Experimental study of critical-mass fluctuations in an evolving sandpile. Phys Rev Lett1990; 65: 1120.
89.
JaegerHMLiuChNagelSR. Relaxation at the angle of repose. Phys Rev Lett1989; 62: 40.
90.
MarchettiMC. Depinning and plasticity of driven disordered lattices. In: Carmen MiguelMRubiM (eds) Jamming, yielding, and irreversible deformation in condensed matter, Lecture Notes in Physics. Berlin and Heidelberg: Springer-Verlag, 2006, pp. 137–157.
91.
DenisovDVLorinczKAWrightWJ, et al. Universal slip dynamics in metallic glasses and granular matter – linking frictional weakening with inertial effects. Scientific Reports2017; 7: 43376.
92.
NicolasABarratJLRottlerJ. Effects of inertia on the steady-shear rheology of disordered solids. Phys Rev Lett2016; 116: 058303.
93.
KarimiKBarratJL. Role of inertia in the rheology of amorphous systems: a finite-element-based elastoplastic model. Phys Rev E2016; 93(2): 022904.
94.
de GeusTWWyartM. Short-range depinning in the presence of velocity-weakening. arXiv [preprint], 2024. DOI: 10.48550/arXiv.2411.06732.
95.
ClancyICorcoranD. Criticality in the Burridge-Knopoff model. Phys Rev E2005; 71(4): 046124.
96.
SalernoKMMaloneyCERobbinsMO. Avalanches in strained amorphous solids: does inertia destroy critical behavior?Phys Rev Lett2012; 109: 105703.
97.
SalernoKMRobbinsMO. Effect of inertia on sheared disordered solids: critical scaling of avalanches in two and three dimensions. Phys Rev E2013; 88: 062206.
98.
KarimiKFerreroEEBarratJL. Inertia and universality of avalanche statistics: the case of slowly deformed amorphous solids. Phys Rev E2017; 95: 013003.
99.
TruskinovskyL. Transition to detonation in dynamic phase changes. Arch Ration Mech Anal1994; 125: 375–397.
100.
ReidAGoodingR. Pattern formation in a 2D elastic solid. Physica A1997; 239(1–3): 1–10.
101.
BalesGGoodingR. Interfacial dynamics at a first-order phase transition involving strain: dynamical twin formation. Phys Rev Lett1991; 67(24): 3412–3415.
102.
ElmerFJ. Avalanches in the weakly driven Frenkel-Kontorova model. Phys Rev E1994; 50(6): 4470.
103.
DingXLookmanTZhaoZ, et al. Dynamically strained ferroelastics: statistical behavior in elastic and plastic regimes. Phys Rev B2013; 87(9): 094109.
104.
AhluwaliaRAnanthakrishnaG. Power-law statistics for avalanches in a martensitic transformation. Phys Rev Lett2001; 86(18): 4076–4079.
105.
SreekalaSAhluwaliaRAnanthakrishnaG. Precursors and power-law statistics of acoustic emission and shape memory effect in martensites. Phys Rev B2004; 70: 224105.
106.
PaulABhattacharyaJSenguptaS, et al. Non-affine deformation in microstructure selection in solids II: elastoplastic theory for thedynamics of solid state transformations. J Phys Condens Matter2008; 20(36): 365211.
107.
BallJMJamesR. Incompatible sets of gradients and metastability. Arch Ration Mech Anal2015; 218: 1363–1416.
108.
BouvilleMAhluwaliaR. Effect of lattice-mismatch-induced strains on coupled diffusive and displacive phase transformations. Phys Rev B2007; 75(5): 054110.
109.
ZhangZJamesRDMüllerS. Energy barriers and hysteresis in martensitic phase transformations. Acta Mater2009; 57(15): 4332–4352.
110.
AtliKFrancoBKaramanI, et al. Influence of crystallographic compatibility on residual strain of TiNi based shape memory alloys during thermo-mechanical cycling. Mater Sci Eng A2013; 574: 9–16.
111.
YangYXuYZhouY, et al. Nonthermoelastic martensitic features in ideal martensites due to volume effects. Phys Rev B2023; 108(2): 024102.
112.
ShenoySRLookmanTSaxenaA, et al. Martensitic textures: multiscale consequences of elastic compatibility. Phys Rev B1999; 60(18): R12537–R12541.
113.
AhluwaliaRLookmanTSaxenaA. Dynamic strain loading of cubic to tetragonal martensites. Acta Mater2006; 54(8): 2109–2120.
114.
KertészJKissL. The noise spectrum in the model of self-organised criticality. J Phys Math General1990; 23(9): L433.
115.
KuntzMCSethnaJP. Noise in disordered systems: the power spectrum and dynamic exponents in avalanche models. Phys Rev B2000; 62: 11699–11708.
116.
LaursonLAlavaMJZapperiS. Power spectra of self-organized critical sandpiles. J Stat Mech2005; 2005(11): L11001.
117.
LaursonLAlavaMJ. 1/ f noise and avalanche scaling in plastic deformation. Phys Rev E2006; 74(6): 066106.
118.
NandiMKSarracinoAHerrmannHJ, et al. Scaling of avalanche shape and activity power spectrum in neuronal networks. Phys Rev E2022; 106(2): 024304.
119.
RuseckasJKaulakysB. Scaling properties of signals as origin of 1/f noise. J Stat Mech2014; 2014(6): P06005.
120.
KaySM. Fundamentals of statistical signal processing: estimation theory. Hoboken, NJ: Prentice-Hall, 1993.
121.
TruskinovskyLVainchteinA. The origin of nucleation peak in transformational plasticity. J Mech Phys Solids2004; 52(6): 1421–1446.
EfendievYRTruskinovskyL. Thermalization of a driven bi-stable FPU chain. Continuum Mech Thermodyn2010; 22(6–8): 679–698.
127.
WestBJShlesingerMF. On the ubiquity of 1/f noise. Int J Mod Phys B1989; 3(6): 795–819.
128.
MilottiE. 1/f noise: a pedagogical review. 2002. DOI: 10.48550/arXiv.physics/0204033.
129.
PolizziSPérez-RecheFJArneodoA, et al. Power-law and log-normal avalanche size statistics in random growth processes. Phys Rev E2021; 104(5): L052101.
130.
SornetteD. Sweeping of an instability: an alternative to self-organized criticality to get powerlaws without parameter tuning. J Phys1994; 4(2): 209–221.
131.
SornetteD. Power laws without parameter tuning: an alternative to self-organized criticality. Phys Rev Lett1994; 72(14): 2306.
132.
VivèsEOrtínJMañosaL, et al. Distribution of acoustic emission avalanches in martensitic transformations. J Phys IV1995; 5(C2): C2–59.
133.
UrbachJSMadisonRCMarkertJT. Interface depinning, self-organized criticality, and the Barkhausen effect. Phys Rev Lett1995; 75(2): 276–279.
134.
DurinGBertottiGMagniA. Fractals, scaling and the question of self-organized criticality in magnetization processes. Fractals1995; 3(2): 351–370.
135.
SchmelzerJUlbrichtH. Thermodynamics of finite systems and the kinetics of first-order phase transitions. J Colloid Interface Sci1987; 117(2): 325–338.
136.
SalmanOUMuiteBFinelA. Origin of stabilization of macrotwin boundaries in martensites. Eur Phys J B2019; 92: 1–9.
137.
CarlsonJMLangerJSShawBE, et al. Intrinsic properties of a Burridge-Knopoff model of an earthquake fault. Phys Rev A1991; 44(2): 884–897.
ElmerFJ. Is self-organized criticality possible in dry friction? In: PerssonBNJTosattiE (eds) Physics of sliding friction. Dordrecht: Springer, 1996, pp. 433–447.
140.
GorbushinNVainchteinATruskinovskyL. Transition fronts and their universality classes. Phys Rev E2022; 106(2): 024210.
141.
PuglisiGTruskinovskyL. Mechanics of a discrete chain with bi-stable elements. J Mech Phys Solids2000; 48(1): 1–27.
142.
PuglisiGTruskinovskyL. A mechanism of transformational plasticity. Contin Mech Thermodyn2002; 14: 437–457.
143.
PuglisiGTruskinovskyL. Rate independent hysteresis in a bi-stable chain. J Mech Phys Solids2002; 50(2): 165–187.
144.
BhattacharyaKFiroozyeNBJamesRD, et al. Restrictions on microstructure. Proc R Soc Edinb A1994; 124(5): 843–878.
145.
SmyshlyaevVWillisJ. On the relation of a three-well energy. Proc R Soc Lond A1999; 455(1983): 779–814.
146.
GrabovskyYTruskinovskyL. A class of nonlinear elasticity problems with no local but many global minimizers. J Elast2023; 154(1): 147–171.
CurnoeSHJacobsAE. Twin wall of proper cubic-tetragonal ferroelastics. Phys Rev B Condens Matter Mater Phys2000; 62(18): R11925–R11928.
149.
BudianskyBTruskinovskyL. On the mechanics of stress-induced phase transformation in zirconia. J Mech Phys Solids1993; 41(9): 1445–1459.
150.
FaddaGTruskinovskyLZanzottoG. Unified Landau description of the tetragonal, orthorhombic, and monoclinic phases of zirconia. Phys Rev B2002; 66(17): 174107.
151.
SalmanOU. Modeling of spatio-temporal dynamics and patterning mechanisms of martensites by phase-field and lagrangian methods. PhD Thesis, Université Pierre et Marie Curie, Paris, 2009.
152.
BallJHolmesPJJamesR, et al. On the dynamics of fine structure. J Nonlinear Sci1991; 1: 17–70.
153.
FrieseckeGMcLeodJ. Dynamic stability of non–minimizing phase mixtures. Proc R Soc Lond A1997; 453(1966): 2427–2436.
154.
CurnoeSHJacobsAE. Time evolution of tetragonal-orthorhombic ferroelastics. Phys Rev B2001; 64(6): 064101.
155.
JacobsAECurnoeSHDesaiRC. Simulations of cubic-tetragonal ferroelastics. Phys Rev B2003; 68(22): 224104.
156.
MutoSOshimaRFujitaFE. Elastic softening and elastic strain energy consideration in the fcc—fct transformation of Fe-Pd alloys. Acta Metall Mater1990; 38(4): 685–694.
157.
KarthaSKrumhanslJASethnaJP, et al. Disorder-driven pretransitional tweed pattern in martensitic transformations. Phys Rev B1995; 52(2): 803–822.
158.
SatoMGrierBHShapiroSM, et al. Effect of magnetic ordering on the lattice dynamics of FCC Fe1-xPdx. J Phys1982; 12(10): 2117–2129.
159.
OshimaRSugiyamaMFujitaFE. Tweed structures associated with Fcc-Fct transformations in Fe-Pd alloys. Metall Trans1988; 19(4): 803–810.
160.
TestardiLRBatemanTB. Lattice instability of high-transition-temperature superconductors. II. Single-CrystalV3Si results. Phys Rev1967; 154(2): 402–410.
161.
RodneyDBouarYLFinelA. Phase field methods and dislocations. Acta Mater2001; 51: 17–30.
162.
PitteriMZanzottoG. Continuum models for phase transitions and twinning in crystals. Boca Raton, FL: Chapman and Hall/CRC, 2002.
163.
ArtemevAJinYKhachaturyanAG. Three-dimensional phase field model and simulation of cubic → tetragonal martensitic transformation in polycrystals. Philos Mag A2002; 82(6): 1249–1270.
164.
MamivandMZaeemMAEl KadiriH. A review on phase field modeling of martensitic phase transformation. Comput Mater Sci2013; 77: 304–311.
165.
NiYJinYMKhachaturyanAG. The transformation sequences in the cubic→tetragonal decomposition. Acta Mater2007; 55(14): 4903–4914.
166.
TurteltaubSSuikerASJ. A multiscale thermomechanical model for cubic to tetragonal martensitic phase transformations. Int J Solids Struct2006; 43(14–15): 4509–4545.
167.
IchitsuboTTanakaKKoiwaM, et al. Kinetics of cubic to tetragonal transformation under external field by the time-dependent Ginzburg-Landau approach. Phys Rev B2000; 62: 5435–5441.
168.
HildebrandFEMieheC. A phase field model for the formation and evolution of martensitic laminate microstructure at finite strains. Philos Mag2012; 92(34): 4250–4290.
169.
IdesmanAChoJLevitasV. Finite element modeling of dynamics of martensitic phase transitions. Appl Phys Lett2008; 93: 043102.
170.
ClausetAShaliziCRNewmanMEJ. Power-law distributions in empirical data. SIAM Rev Soc Ind Appl Math2009; 51(4): 661–703.
171.
VivesEGoicoecheaJOrtínJ, et al. Universality in models for disorder-induced phase transitions. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics1995; 52(1): R5–R8.
172.
LeVequeRJ. Finite difference methods for ordinary and partial differential equations, steady state and time dependent problems. Philadelphia, PA: SIAM, 2007.
